A method of convergence acceleration
of some continued fractions II
 

Initial coefficients of the asymptotic expansion of CF tails 

 

Rafal Nowak 

Institute of Computer Science, University of Wroclaw, ul. F. Joliot-Curie 15, 50-383 Wroclaw, Poland 

 

> restart;
 

Two-variant continued fraction (CF) 

We consider two-variant CF of the form 

Typesetting:-mrow(Typesetting:-mi( 

We truncate all the asymptotic expansions after Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi( 

> MAX := 7;
 

7 (1.1)
 

> a := unapply( add( p[j]/n^j , j=-2..MAX ), n );        # a_n
b := unapply( add( q[j]/n^j , j=-1..MAX ), n );        # b_n

`a'` := unapply( add( `p'`[j]/n^j , j=-2..MAX ), n );  # a_n'
`b'` := unapply( add( `q'`[j]/n^j , j=-1..MAX ), n );  # b_n'
 

Asymptotic expansions of numerators and denominators 

 

 

 

proc (n) options operator, arrow; `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`...
proc (n) options operator, arrow; `+`(`*`(q[-1], `*`(n)), q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `...
proc (n) options operator, arrow; `+`(`*`(`p'`[-2], `*`(`^`(n, 2))), `*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/...
proc (n) options operator, arrow; `+`(`*`(`q'`[-1], `*`(n)), `q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4)))... (1.2)
 

Asymptotic expansions of tails 

> EQ := '(`b'`(n)*b(n+1) + a(n+1))*X(n) + `b'`(n)*X(n)*X(n+1) - `a'`(n)*X(n+1) - `a'`(n)*b(n+1)' = 0;
# EQ := '(`b'`(n)*b(n) + a(n))*X(n) + `b'`(n)*X(n)*X(n+1) - `a'`(n)*X(n+1) - `a'`(n)*b(n)' = 0;
 

Bilinear equation satisfied by tails of CF 

`+`(`*`(`+`(`*`(`b'`(n), `*`(b(`+`(n, 1)))), a(`+`(n, 1))), `*`(X(n))), `*`(`b'`(n), `*`(X(n), `*`(X(`+`(n, 1))))), `-`(`*`(`a'`(n), `*`(X(`+`(n, 1))))), `-`(`*`(`a'`(n), `*`(b(`+`(n, 1)))))) = 0
`+`(`*`(`+`(`*`(`b'`(n), `*`(b(`+`(n, 1)))), a(`+`(n, 1))), `*`(X(n))), `*`(`b'`(n), `*`(X(n), `*`(X(`+`(n, 1))))), `-`(`*`(`a'`(n), `*`(X(`+`(n, 1))))), `-`(`*`(`a'`(n), `*`(b(`+`(n, 1)))))) = 0
(2.1)
 

> X := unapply( add( tau[j]/n^(j/2) , j=-4..MAX ), n );    # X_n
 

Asymptotic expansion of the solutions of (2.1). 

proc (n) options operator, arrow; `+`(`*`(tau[-4], `*`(`^`(n, 2))), `*`(tau[-3], `*`(`^`(n, `/`(3, 2)))), `*`(tau[-2], `*`(n)), `*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n...
proc (n) options operator, arrow; `+`(`*`(tau[-4], `*`(`^`(n, 2))), `*`(tau[-3], `*`(`^`(n, `/`(3, 2)))), `*`(tau[-2], `*`(n)), `*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n...
(2.2)
 

We calculate the coefficients Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(of Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi( in the left hand side of equation (2.1). 

Observe that LHS of (2.1) is at most Typesetting:-mrow(Typesetting:-mi(, so we calculate Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(for Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

> # c(m) = coefficient of n^(-m/2) in the left hand side of (2.1)

EQ2 := simplify( subs( n=N^2, EQ ) ) assuming N>0:         # Warning: this can take some time
LHS := convert( asympt( lhs(EQ2), N, 10*MAX ), polynom ):  # Warning: this can take some time
c := m -> coeff( LHS, N, -m );
 

proc (m) options operator, arrow; coeff(LHS, N, `+`(`-`(m))) end proc (2.3)
 

The system (*) of equations satisfied by coefficients Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

> for m from -10 to -2 do print( c(m)=0 ); end: m := 'm':
 

 

 

 

 

 

 

 

 

`*`(`q'`[-1], `*`(`^`(tau[-4], 2))) = 0
`+`(`*`(2, `*`(`q'`[-1], `*`(tau[-3], `*`(tau[-4]))))) = 0
`+`(`*`(2, `*`(`q'`[-1], `*`(`^`(tau[-4], 2)))), `*`(`q'`[-1], `*`(q[-1], `*`(tau[-4]))), `*`(2, `*`(`q'`[-1], `*`(tau[-2], `*`(tau[-4])))), `*`(`q'`[0], `*`(`^`(tau[-4], 2))), `-`(`*`(`p'`[-2], `*`(t...
`+`(`*`(`/`(7, 2), `*`(`q'`[-1], `*`(tau[-3], `*`(tau[-4])))), `*`(p[-2], `*`(tau[-3])), `*`(`q'`[-1], `*`(q[-1], `*`(tau[-3]))), `*`(2, `*`(`q'`[0], `*`(tau[-3], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1],...
`+`(`*`(`/`(7, 2), `*`(`q'`[-1], `*`(tau[-3], `*`(tau[-4])))), `*`(p[-2], `*`(tau[-3])), `*`(`q'`[-1], `*`(q[-1], `*`(tau[-3]))), `*`(2, `*`(`q'`[0], `*`(tau[-3], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1],...
`+`(`*`(2, `*`(`q'`[-1], `*`(tau[0], `*`(tau[-4])))), `*`(2, `*`(`q'`[0], `*`(tau[-2], `*`(tau[-4])))), `*`(`q'`[1], `*`(`^`(tau[-4], 2))), `-`(`*`(`p'`[-2], `*`(q[-1]))), `*`(`q'`[-1], `*`(`^`(tau[-4...
`+`(`*`(2, `*`(`q'`[-1], `*`(tau[0], `*`(tau[-4])))), `*`(2, `*`(`q'`[0], `*`(tau[-2], `*`(tau[-4])))), `*`(`q'`[1], `*`(`^`(tau[-4], 2))), `-`(`*`(`p'`[-2], `*`(q[-1]))), `*`(`q'`[-1], `*`(`^`(tau[-4...
`+`(`*`(2, `*`(`q'`[-1], `*`(tau[0], `*`(tau[-4])))), `*`(2, `*`(`q'`[0], `*`(tau[-2], `*`(tau[-4])))), `*`(`q'`[1], `*`(`^`(tau[-4], 2))), `-`(`*`(`p'`[-2], `*`(q[-1]))), `*`(`q'`[-1], `*`(`^`(tau[-4...
`+`(`*`(`/`(5, 2), `*`(`q'`[-1], `*`(tau[-1], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1], `*`(tau[1], `*`(tau[-4])))), `*`(`/`(7, 2), `*`(`q'`[0], `*`(tau[-3], `*`(tau[-4])))), `*`(2, `*`(`q'`[0], `*`(tau[-...
`+`(`*`(`/`(5, 2), `*`(`q'`[-1], `*`(tau[-1], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1], `*`(tau[1], `*`(tau[-4])))), `*`(`/`(7, 2), `*`(`q'`[0], `*`(tau[-3], `*`(tau[-4])))), `*`(2, `*`(`q'`[0], `*`(tau[-...
`+`(`*`(`/`(5, 2), `*`(`q'`[-1], `*`(tau[-1], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1], `*`(tau[1], `*`(tau[-4])))), `*`(`/`(7, 2), `*`(`q'`[0], `*`(tau[-3], `*`(tau[-4])))), `*`(2, `*`(`q'`[0], `*`(tau[-...
`+`(`*`(`/`(5, 2), `*`(`q'`[-1], `*`(tau[-1], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1], `*`(tau[1], `*`(tau[-4])))), `*`(`/`(7, 2), `*`(`q'`[0], `*`(tau[-3], `*`(tau[-4])))), `*`(2, `*`(`q'`[0], `*`(tau[-...
`+`(`*`(2, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-4]))), `*`(`q'`[-1], `*`(`^`(tau[-1], 2))), `*`(2, `*`(`q'`[-1], `*`(tau[2], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1],...
`+`(`*`(2, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-4]))), `*`(`q'`[-1], `*`(`^`(tau[-1], 2))), `*`(2, `*`(`q'`[-1], `*`(tau[2], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1],...
`+`(`*`(2, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-4]))), `*`(`q'`[-1], `*`(`^`(tau[-1], 2))), `*`(2, `*`(`q'`[-1], `*`(tau[2], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1],...
`+`(`*`(2, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-4]))), `*`(`q'`[-1], `*`(`^`(tau[-1], 2))), `*`(2, `*`(`q'`[-1], `*`(tau[2], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1],...
`+`(`*`(2, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-4]))), `*`(`q'`[-1], `*`(`^`(tau[-1], 2))), `*`(2, `*`(`q'`[-1], `*`(tau[2], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1],...
`+`(`*`(2, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-4]))), `*`(`q'`[-1], `*`(`^`(tau[-1], 2))), `*`(2, `*`(`q'`[-1], `*`(tau[2], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1],...
`+`(`*`(`q'`[0], `*`(q[0], `*`(tau[-3]))), `*`(`/`(7, 8), `*`(`q'`[-1], `*`(tau[-1], `*`(tau[-4])))), `*`(`/`(3, 2), `*`(`q'`[-1], `*`(tau[1], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1], `*`(tau[1], `*`(tau...
`+`(`*`(`q'`[0], `*`(q[0], `*`(tau[-3]))), `*`(`/`(7, 8), `*`(`q'`[-1], `*`(tau[-1], `*`(tau[-4])))), `*`(`/`(3, 2), `*`(`q'`[-1], `*`(tau[1], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1], `*`(tau[1], `*`(tau...
`+`(`*`(`q'`[0], `*`(q[0], `*`(tau[-3]))), `*`(`/`(7, 8), `*`(`q'`[-1], `*`(tau[-1], `*`(tau[-4])))), `*`(`/`(3, 2), `*`(`q'`[-1], `*`(tau[1], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1], `*`(tau[1], `*`(tau...
`+`(`*`(`q'`[0], `*`(q[0], `*`(tau[-3]))), `*`(`/`(7, 8), `*`(`q'`[-1], `*`(tau[-1], `*`(tau[-4])))), `*`(`/`(3, 2), `*`(`q'`[-1], `*`(tau[1], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1], `*`(tau[1], `*`(tau...
`+`(`*`(`q'`[0], `*`(q[0], `*`(tau[-3]))), `*`(`/`(7, 8), `*`(`q'`[-1], `*`(tau[-1], `*`(tau[-4])))), `*`(`/`(3, 2), `*`(`q'`[-1], `*`(tau[1], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1], `*`(tau[1], `*`(tau...
`+`(`*`(`q'`[0], `*`(q[0], `*`(tau[-3]))), `*`(`/`(7, 8), `*`(`q'`[-1], `*`(tau[-1], `*`(tau[-4])))), `*`(`/`(3, 2), `*`(`q'`[-1], `*`(tau[1], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1], `*`(tau[1], `*`(tau...
`+`(`*`(`q'`[0], `*`(q[0], `*`(tau[-3]))), `*`(`/`(7, 8), `*`(`q'`[-1], `*`(tau[-1], `*`(tau[-4])))), `*`(`/`(3, 2), `*`(`q'`[-1], `*`(tau[1], `*`(tau[-4])))), `*`(2, `*`(`q'`[-1], `*`(tau[1], `*`(tau...
`+`(`*`(3, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-2]))), `*`(`/`(1, 2), `*`(`q'`[-1], `*`(`^`(tau[-1], 2)))), `*`(`q'`[0], `*`(`^`(tau[-1], 2))), `*`(p[1], `*`(tau[...
`+`(`*`(3, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-2]))), `*`(`/`(1, 2), `*`(`q'`[-1], `*`(`^`(tau[-1], 2)))), `*`(`q'`[0], `*`(`^`(tau[-1], 2))), `*`(p[1], `*`(tau[...
`+`(`*`(3, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-2]))), `*`(`/`(1, 2), `*`(`q'`[-1], `*`(`^`(tau[-1], 2)))), `*`(`q'`[0], `*`(`^`(tau[-1], 2))), `*`(p[1], `*`(tau[...
`+`(`*`(3, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-2]))), `*`(`/`(1, 2), `*`(`q'`[-1], `*`(`^`(tau[-1], 2)))), `*`(`q'`[0], `*`(`^`(tau[-1], 2))), `*`(p[1], `*`(tau[...
`+`(`*`(3, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-2]))), `*`(`/`(1, 2), `*`(`q'`[-1], `*`(`^`(tau[-1], 2)))), `*`(`q'`[0], `*`(`^`(tau[-1], 2))), `*`(p[1], `*`(tau[...
`+`(`*`(3, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-2]))), `*`(`/`(1, 2), `*`(`q'`[-1], `*`(`^`(tau[-1], 2)))), `*`(`q'`[0], `*`(`^`(tau[-1], 2))), `*`(p[1], `*`(tau[...
`+`(`*`(3, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-2]))), `*`(`/`(1, 2), `*`(`q'`[-1], `*`(`^`(tau[-1], 2)))), `*`(`q'`[0], `*`(`^`(tau[-1], 2))), `*`(p[1], `*`(tau[...
`+`(`*`(3, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-2]))), `*`(`/`(1, 2), `*`(`q'`[-1], `*`(`^`(tau[-1], 2)))), `*`(`q'`[0], `*`(`^`(tau[-1], 2))), `*`(p[1], `*`(tau[...
`+`(`*`(3, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-2]))), `*`(`/`(1, 2), `*`(`q'`[-1], `*`(`^`(tau[-1], 2)))), `*`(`q'`[0], `*`(`^`(tau[-1], 2))), `*`(p[1], `*`(tau[...
`+`(`*`(3, `*`(`q'`[1], `*`(tau[-4], `*`(tau[-2])))), `*`(`q'`[0], `*`(q[0], `*`(tau[-2]))), `*`(`/`(1, 2), `*`(`q'`[-1], `*`(`^`(tau[-1], 2)))), `*`(`q'`[0], `*`(`^`(tau[-1], 2))), `*`(p[1], `*`(tau[...
(2.1.1)
 

Initial coefficients Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

Class D_{10} (p[-1] = p'[-1])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := 0; p[-2] := 0; `p'`[-1] := p[-1]; q[-1] := 0; `q'`[-1] := 0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

 

 

 

0
0
p[-1]
0
0
a(n) = `+`(`*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), `*`(`^`(n, 5))), `/`(`*`(p[6...
`a'`(n) = `+`(`*`(p[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, 4))), `/`(`*`(`p'`[5]), `*`(`^`...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `/`(`*`(q[6]), `*`(`^`(n, 6))),...
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `/`(`*`(`q... (3.1.1)
 

> c(-10)=0; c(-9)=0;
 

First two equations of the system (*) are satisfied trivially 

 

0 = 0
0 = 0 (3.1.2)
 

> c(-8)=0; tau[-4] := 0;
 

First nontrivial equation. 

 

`*`(`q'`[0], `*`(`^`(tau[-4], 2))) = 0
0 (3.1.3)
 

> c(-7)=0;
 

0 = 0 (3.1.4)
 

> c(-6)=0; tau[-3] := 0;
 

 

`*`(`q'`[0], `*`(`^`(tau[-3], 2))) = 0
0 (3.1.5)
 

> c(-5)=0;
 

0 = 0 (3.1.6)
 

> c(-4)=0; tau[-2] := 0;
 

 

`*`(`q'`[0], `*`(`^`(tau[-2], 2))) = 0
0 (3.1.7)
 

> c(-3);
 

0 (3.1.8)
 

Initial coefficient Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

> c(-2); tau[-1] = solve( %, tau[-1] );
 

> mu := 1; k := -2:
 

 

 

`+`(`*`(`q'`[0], `*`(`^`(tau[-1], 2))), `-`(`*`(p[-1], `*`(q[0]))))
tau[-1] = (`/`(`*`(`^`(`*`(`q'`[0], `*`(p[-1], `*`(q[0]))), `/`(1, 2))), `*`(`q'`[0])), `+`(`-`(`/`(`*`(`^`(`*`(`q'`[0], `*`(p[-1], `*`(q[0]))), `/`(1, 2))), `*`(`q'`[0])))))
1 (3.1.1.1)
 

> alpha = coeff( c(k) , tau[-mu], 2 );
beta  = coeff( c(k) , tau[-mu], 1 );
gamma = coeff( c(k) , tau[-mu], 0 );
 

Quadratic equation satisfied by the coefficient Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

 

 

alpha = `q'`[0]
beta = 0
gamma = `+`(`-`(`*`(p[-1], `*`(q[0])))) (3.1.1.2)
 

The next coefficients 

Choice of the first solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[1];
eq := [%]: for j from 1 to 3 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); print(%): eq := [ op(eq), % ]: end:
 

 

 

 

tau[-1] = `/`(`*`(`^`(`*`(`q'`[0], `*`(p[-1], `*`(q[0]))), `/`(1, 2))), `*`(`q'`[0]))
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 4), `*`(`+`(`-`(`*`(2, `*`(`p'`[0]))), `*`(2, `*`(`q'`[0], `*`(q[0]))), p[-1], `*`(2, `*`(p[0]))))), `*`(`q'`[0]))))
tau[1] = `+`(`/`(`*`(`/`(1, 32), `*`(`+`(`*`(4, `*`(`^`(`q'`[0], 2), `*`(`^`(q[0], 2)))), `-`(`*`(8, `*`(p[-1], `*`(`p'`[0])))), `*`(3, `*`(`^`(p[-1], 2))), `*`(8, `*`(p[-1], `*`(p[0]))), `-`(`*`(8, `...
tau[1] = `+`(`/`(`*`(`/`(1, 32), `*`(`+`(`*`(4, `*`(`^`(`q'`[0], 2), `*`(`^`(q[0], 2)))), `-`(`*`(8, `*`(p[-1], `*`(`p'`[0])))), `*`(3, `*`(`^`(p[-1], 2))), `*`(8, `*`(p[-1], `*`(p[0]))), `-`(`*`(8, `...
tau[1] = `+`(`/`(`*`(`/`(1, 32), `*`(`+`(`*`(4, `*`(`^`(`q'`[0], 2), `*`(`^`(q[0], 2)))), `-`(`*`(8, `*`(p[-1], `*`(`p'`[0])))), `*`(3, `*`(`^`(p[-1], 2))), `*`(8, `*`(p[-1], `*`(p[0]))), `-`(`*`(8, `...
tau[2] = `+`(`-`(`/`(`*`(`/`(1, 64), `*`(`+`(`-`(`*`(8, `*`(p[0], `*`(`p'`[0])))), `-`(`*`(32, `*`(`q'`[1], `*`(p[-1], `*`(q[0]))))), `*`(3, `*`(`^`(p[-1], 2))), `*`(32, `*`(q[1], `*`(`^`(`q'`[0], 2),...
tau[2] = `+`(`-`(`/`(`*`(`/`(1, 64), `*`(`+`(`-`(`*`(8, `*`(p[0], `*`(`p'`[0])))), `-`(`*`(32, `*`(`q'`[1], `*`(p[-1], `*`(q[0]))))), `*`(3, `*`(`^`(p[-1], 2))), `*`(32, `*`(q[1], `*`(`^`(`q'`[0], 2),...
tau[2] = `+`(`-`(`/`(`*`(`/`(1, 64), `*`(`+`(`-`(`*`(8, `*`(p[0], `*`(`p'`[0])))), `-`(`*`(32, `*`(`q'`[1], `*`(p[-1], `*`(q[0]))))), `*`(3, `*`(`^`(p[-1], 2))), `*`(32, `*`(q[1], `*`(`^`(`q'`[0], 2),...
(3.1.2.1.1)
 

Choice of the second solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[2];
 

> eq := [%]: for j from 1 to 3 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] );  print(%): eq := [ op(eq), % ]: end:
 

 

 

 

tau[-1] = `+`(`-`(`/`(`*`(`^`(`*`(`q'`[0], `*`(p[-1], `*`(q[0]))), `/`(1, 2))), `*`(`q'`[0]))))
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 4), `*`(`+`(`-`(`*`(2, `*`(`p'`[0]))), `*`(2, `*`(`q'`[0], `*`(q[0]))), p[-1], `*`(2, `*`(p[0]))))), `*`(`q'`[0]))))
tau[1] = `+`(`-`(`/`(`*`(`/`(1, 32), `*`(`+`(`*`(4, `*`(`^`(`q'`[0], 2), `*`(`^`(q[0], 2)))), `-`(`*`(8, `*`(p[-1], `*`(`p'`[0])))), `*`(3, `*`(`^`(p[-1], 2))), `*`(8, `*`(p[-1], `*`(p[0]))), `-`(`*`(...
tau[1] = `+`(`-`(`/`(`*`(`/`(1, 32), `*`(`+`(`*`(4, `*`(`^`(`q'`[0], 2), `*`(`^`(q[0], 2)))), `-`(`*`(8, `*`(p[-1], `*`(`p'`[0])))), `*`(3, `*`(`^`(p[-1], 2))), `*`(8, `*`(p[-1], `*`(p[0]))), `-`(`*`(...
tau[1] = `+`(`-`(`/`(`*`(`/`(1, 32), `*`(`+`(`*`(4, `*`(`^`(`q'`[0], 2), `*`(`^`(q[0], 2)))), `-`(`*`(8, `*`(p[-1], `*`(`p'`[0])))), `*`(3, `*`(`^`(p[-1], 2))), `*`(8, `*`(p[-1], `*`(p[0]))), `-`(`*`(...
tau[2] = `+`(`-`(`/`(`*`(`/`(1, 64), `*`(`+`(`-`(`*`(8, `*`(p[0], `*`(`p'`[0])))), `-`(`*`(32, `*`(`q'`[1], `*`(p[-1], `*`(q[0]))))), `*`(3, `*`(`^`(p[-1], 2))), `*`(32, `*`(q[1], `*`(`^`(`q'`[0], 2),...
tau[2] = `+`(`-`(`/`(`*`(`/`(1, 64), `*`(`+`(`-`(`*`(8, `*`(p[0], `*`(`p'`[0])))), `-`(`*`(32, `*`(`q'`[1], `*`(p[-1], `*`(q[0]))))), `*`(3, `*`(`^`(p[-1], 2))), `*`(32, `*`(q[1], `*`(`^`(`q'`[0], 2),...
tau[2] = `+`(`-`(`/`(`*`(`/`(1, 64), `*`(`+`(`-`(`*`(8, `*`(p[0], `*`(`p'`[0])))), `-`(`*`(32, `*`(`q'`[1], `*`(p[-1], `*`(q[0]))))), `*`(3, `*`(`^`(p[-1], 2))), `*`(32, `*`(q[1], `*`(`^`(`q'`[0], 2),...
(3.1.2.2.1)
 

Class D_{10} (p[-1] ≠ p'[-1])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := 0; p[-2] := 0; q[-1] := 0; `q'`[-1] :=  0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

 

 

0
0
0
0
a(n) = `+`(`*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), `*`(`^`(n, 5))), `...`)
`a'`(n) = `+`(`*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, 4))), `/`(`*`(`p'`[5]), `*`(...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `...`) (3.2.1)
 

> c(-10)=0; c(-9)=0;
 

 

0 = 0
0 = 0 (3.2.2)
 

> c(-8)=0; tau[-4] := 0;
 

 

`*`(`q'`[0], `*`(`^`(tau[-4], 2))) = 0
0 (3.2.3)
 

> c(-7)=0;
 

0 = 0 (3.2.4)
 

> c(-6)=0; tau[-3] := 0;
 

 

`*`(`q'`[0], `*`(`^`(tau[-3], 2))) = 0
0 (3.2.5)
 

> c(-5)=0;
 

0 = 0 (3.2.6)
 

Initial coefficient tau[-mu] 

> mu := 2; k := -4:
c(k); tau[-mu] = solve( %, tau[-mu] );
 

 

 

2
`+`(`-`(`*`(`p'`[-1], `*`(tau[-2]))), `*`(p[-1], `*`(tau[-2])), `*`(`q'`[0], `*`(`^`(tau[-2], 2))))
tau[-2] = (0, `+`(`-`(`/`(`*`(`+`(`-`(`p'`[-1]), p[-1])), `*`(`q'`[0]))))) (3.2.1.1)
 

> alpha = coeff( c(k) , tau[-mu], 2 );
beta  = coeff( c(k) , tau[-mu], 1 );
gamma = coeff( c(k) , tau[-mu], 0 );
 

Quadratic equation satisfied by the coefficient Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

 

 

alpha = `q'`[0]
beta = `+`(`-`(`p'`[-1]), p[-1])
gamma = 0 (3.2.1.2)
 

The next coefficients 

Choice of the first solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[1];
eq := [%]: for j from 1 to 5 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); if j<=7 or j mod 2 = 1 then print(%): end: eq := [ op(eq), % ]: end:
 

 

 

 

 

 

tau[-2] = 0
tau[-1] = 0
tau[0] = `/`(`*`(`p'`[-1], `*`(q[0])), `*`(`+`(`-`(`p'`[-1]), p[-1])))
tau[1] = 0
tau[2] = `+`(`-`(`/`(`*`(`+`(`-`(`*`(p[0], `*`(`^`(`p'`[-1], 2), `*`(q[0])))), `*`(p[0], `*`(`p'`[-1], `*`(q[0], `*`(p[-1])))), `*`(`q'`[0], `*`(`^`(q[0], 2), `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(p[-1...
tau[2] = `+`(`-`(`/`(`*`(`+`(`-`(`*`(p[0], `*`(`^`(`p'`[-1], 2), `*`(q[0])))), `*`(p[0], `*`(`p'`[-1], `*`(q[0], `*`(p[-1])))), `*`(`q'`[0], `*`(`^`(q[0], 2), `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(p[-1...
tau[2] = `+`(`-`(`/`(`*`(`+`(`-`(`*`(p[0], `*`(`^`(`p'`[-1], 2), `*`(q[0])))), `*`(p[0], `*`(`p'`[-1], `*`(q[0], `*`(p[-1])))), `*`(`q'`[0], `*`(`^`(q[0], 2), `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(p[-1...
tau[3] = 0 (3.2.2.1.1)
 

Choice of the second solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[2];
eq := [%]: for j from 1 to 5 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); if j<=7 or j mod 2 = 1 then print(%): end: eq := [ op(eq), % ]: end:
 

 

 

 

 

 

tau[-2] = `+`(`-`(`/`(`*`(`+`(`-`(`p'`[-1]), p[-1])), `*`(`q'`[0]))))
tau[-1] = 0
tau[0] = `/`(`*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `-`(`*`(2, `*`(`q'`[1], `*`(`p'`[-1], `*`(p[-1]))))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `-`(`*`(q[0], `*`(`^`(`q'`[0], 2), `*`(p[-1])))), `-`(...
tau[0] = `/`(`*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `-`(`*`(2, `*`(`q'`[1], `*`(`p'`[-1], `*`(p[-1]))))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `-`(`*`(q[0], `*`(`^`(`q'`[0], 2), `*`(p[-1])))), `-`(...
tau[1] = 0
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[2] = `/`(`*`(`+`(`-`(`*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 4)))), `-`(`*`(`^`(`q'`[1], 2), `*`(`^`(p[-1], 4)))), `*`(4, `*`(`^`(`q'`[1], 2), `*`(`^`(`p'`[-1], 3), `*`(p[-1])))), `-`(`*`(6, `*`(`^...
tau[3] = 0 (3.2.2.2.1)
 

Class D_{11} 

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := 0; p[-2] := 0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

0
0
a(n) = `+`(`*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), `*`(`^`(n, 5))), `...`)
`a'`(n) = `+`(`*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, 4))), `/`(`*`(`p'`[5]), `*`(...
b(n) = `+`(`*`(q[-1], `*`(n)), q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`*`(`q'`[-1], `*`(n)), `q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(... (3.3.1)
 

> c(-10)=0; tau[-4] := 0;
 

 

`*`(`q'`[-1], `*`(`^`(tau[-4], 2))) = 0
0 (3.3.2)
 

> c(-9)=0;
 

0 = 0 (3.3.3)
 

> c(-8)=0; tau[-3] := 0;
 

 

`*`(`q'`[-1], `*`(`^`(tau[-3], 2))) = 0
0 (3.3.4)
 

> c(-7)=0;
 

0 = 0 (3.3.5)
 

Initial coefficient tau[-mu] 

> mu := 2; k := -6:
c(k); tau[-mu] = solve( %, tau[-mu] );
 

 

 

2
`+`(`*`(`q'`[-1], `*`(q[-1], `*`(tau[-2]))), `*`(`q'`[-1], `*`(`^`(tau[-2], 2))))
tau[-2] = (0, `+`(`-`(q[-1]))) (3.3.1.1)
 

> alpha = coeff( c(k) , tau[-mu], 2 );
beta  = coeff( c(k) , tau[-mu], 1 );
gamma = coeff( c(k) , tau[-mu], 0 );
 

Quadratic equation satisfied by the coefficient Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

 

 

alpha = `q'`[-1]
beta = `*`(`q'`[-1], `*`(q[-1]))
gamma = 0 (3.3.1.2)
 

The next coefficients 

Choice of the first solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[1];
eq := [%]: for j from 1 to 7 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); if j<=7 or j mod 2 = 1 then print(%): end: eq := [ op(eq), % ]: end:
 

 

 

 

 

 

 

 

tau[-2] = 0
tau[-1] = 0
tau[0] = `/`(`*`(`p'`[-1]), `*`(`q'`[-1]))
tau[1] = 0
tau[2] = `+`(`-`(`/`(`*`(`+`(`*`(`q'`[0], `*`(q[-1], `*`(`p'`[-1]))), `*`(`p'`[-1], `*`(p[-1])), `-`(`*`(`p'`[0], `*`(q[-1], `*`(`q'`[-1])))))), `*`(`^`(`q'`[-1], 2), `*`(q[-1])))))
tau[3] = 0
tau[4] = `/`(`*`(`+`(`-`(`*`(`q'`[1], `*`(`^`(q[-1], 2), `*`(`p'`[-1], `*`(`q'`[-1]))))), `*`(2, `*`(p[-1], `*`(`q'`[0], `*`(q[-1], `*`(`p'`[-1]))))), `-`(`*`(p[-1], `*`(`p'`[0], `*`(q[-1], `*`(`q'`[-...
tau[4] = `/`(`*`(`+`(`-`(`*`(`q'`[1], `*`(`^`(q[-1], 2), `*`(`p'`[-1], `*`(`q'`[-1]))))), `*`(2, `*`(p[-1], `*`(`q'`[0], `*`(q[-1], `*`(`p'`[-1]))))), `-`(`*`(p[-1], `*`(`p'`[0], `*`(q[-1], `*`(`q'`[-...
tau[4] = `/`(`*`(`+`(`-`(`*`(`q'`[1], `*`(`^`(q[-1], 2), `*`(`p'`[-1], `*`(`q'`[-1]))))), `*`(2, `*`(p[-1], `*`(`q'`[0], `*`(q[-1], `*`(`p'`[-1]))))), `-`(`*`(p[-1], `*`(`p'`[0], `*`(q[-1], `*`(`q'`[-...
tau[5] = 0 (3.3.2.1.1)
 

Choice of the second solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[2];
eq := [%]: for j from 1 to 7 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); if j<=7 or j mod 2 = 1 then print(%): end: eq := [ op(eq), % ]: end:
 

 

 

 

 

 

 

 

tau[-2] = `+`(`-`(q[-1]))
tau[-1] = 0
tau[0] = `+`(`-`(`/`(`*`(`+`(p[-1], `*`(`q'`[-1], `*`(q[0])))), `*`(`q'`[-1]))))
tau[1] = 0
tau[2] = `/`(`*`(`+`(`-`(`*`(p[-1], `*`(q[-1], `*`(`q'`[-1])))), `*`(`q'`[0], `*`(q[-1], `*`(p[-1]))), `-`(`*`(p[0], `*`(q[-1], `*`(`q'`[-1])))), `*`(`p'`[-1], `*`(p[-1])), `-`(`*`(`^`(`q'`[-1], 2), `...
tau[3] = 0
tau[4] = `+`(`-`(`/`(`*`(`+`(`*`(2, `*`(p[-1], `*`(`q'`[0], `*`(q[-1], `*`(`p'`[-1]))))), `-`(`*`(p[-1], `*`(`p'`[0], `*`(q[-1], `*`(`q'`[-1]))))), `*`(`q'`[-1], `*`(q[0], `*`(`p'`[-1], `*`(p[-1])))),...
tau[4] = `+`(`-`(`/`(`*`(`+`(`*`(2, `*`(p[-1], `*`(`q'`[0], `*`(q[-1], `*`(`p'`[-1]))))), `-`(`*`(p[-1], `*`(`p'`[0], `*`(q[-1], `*`(`q'`[-1]))))), `*`(`q'`[-1], `*`(q[0], `*`(`p'`[-1], `*`(p[-1])))),...
tau[4] = `+`(`-`(`/`(`*`(`+`(`*`(2, `*`(p[-1], `*`(`q'`[0], `*`(q[-1], `*`(`p'`[-1]))))), `-`(`*`(p[-1], `*`(`p'`[0], `*`(q[-1], `*`(`q'`[-1]))))), `*`(`q'`[-1], `*`(q[0], `*`(`p'`[-1], `*`(p[-1])))),...
tau[4] = `+`(`-`(`/`(`*`(`+`(`*`(2, `*`(p[-1], `*`(`q'`[0], `*`(q[-1], `*`(`p'`[-1]))))), `-`(`*`(p[-1], `*`(`p'`[0], `*`(q[-1], `*`(`q'`[-1]))))), `*`(`q'`[-1], `*`(q[0], `*`(`p'`[-1], `*`(p[-1])))),...
tau[5] = 0 (3.3.2.2.1)
 

Class D_{20} (p[-2] = p'[-2])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := p[-2]; q[-1] := 0; `q'`[-1] :=  0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

 

p[-2]
0
0
a(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), ...
`a'`(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, ...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `...`) (3.4.1)
 

> c(-10)=0; c(-9)=0;
 

 

0 = 0
0 = 0 (3.4.2)
 

> c(-8)=0; tau[-4] := 0;
 

 

`*`(`q'`[0], `*`(`^`(tau[-4], 2))) = 0
0 (3.4.3)
 

> c(-7)=0;
 

0 = 0 (3.4.4)
 

> c(-6)=0; tau[-3] := 0;
 

 

`*`(`q'`[0], `*`(`^`(tau[-3], 2))) = 0
0 (3.4.5)
 

> c(-5)=0;
 

0 = 0 (3.4.6)
 

Initial coefficient tau[-mu] 

> mu := 2; k := -4:
c(k)=0; tau[-mu] = solve( %, tau[-mu] );
 

 

 

2
`+`(`-`(`*`(`p'`[-1], `*`(tau[-2]))), `*`(p[-1], `*`(tau[-2])), `*`(`q'`[0], `*`(`^`(tau[-2], 2))), `-`(`*`(p[-2], `*`(q[0]))), `*`(p[-2], `*`(tau[-2]))) = 0
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-1], `-`(p[-1]), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-1], `-`(p[-1]), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-1], `-`(p[-1]), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-1], `-`(p[-1]), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p...
(3.4.1.1)
 

> alpha = coeff( c(k) , tau[-mu], 2 );
beta  = coeff( c(k) , tau[-mu], 1 );
gamma = coeff( c(k) , tau[-mu], 0 );
 

Quadratic equation satisfied by the coefficient Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

 

 

alpha = `q'`[0]
beta = `+`(`-`(`p'`[-1]), p[-1], p[-2])
gamma = `+`(`-`(`*`(p[-2], `*`(q[0])))) (3.4.1.2)
 

The next coefficients 

Choice of the first solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[1];
eq := [%]: for j from 1 to 7 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); if j<=3 or j mod 2 = 1 then print(%): end: eq := [ op(eq), % ]: end:
 

 

 

 

 

 

tau[-2] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-1], `-`(p[-1]), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p[...
tau[-2] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-1], `-`(p[-1]), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p[...
tau[-2] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-1], `-`(p[-1]), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p[...
tau[-1] = 0
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(2, `*`(p[-2], `*`(q[1], `*`(`^`(`q'`[0], 2))))), `*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2)))), `-`(`*`(`p'`[0], `*`(`q'`[0], `*`(p[-2])))), `-`(`*`(`p'`[0...
tau[1] = 0
tau[3] = 0
tau[5] = 0 (3.4.2.1.1)
 

Choice of the second solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[2];
eq := [%]: for j from 1 to 7 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); if j<=3 or j mod 2 = 1 then print(%): end: eq := [ op(eq), % ]: end:
 

 

 

 

 

 

tau[-2] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`p'`[-1]), p[-1], p[-2], `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p[-...
tau[-2] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`p'`[-1]), p[-1], p[-2], `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p[-...
tau[-2] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`p'`[-1]), p[-1], p[-2], `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p[-...
tau[-1] = 0
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[0] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-1], 2))), `*`(`q'`[1], `*`(`^`(p[-2], 2))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`^`(`q'`[0], 2))))), `-...
tau[1] = 0
tau[3] = 0
tau[5] = 0 (3.4.2.2.1)
 

Class D_{20} (p[-2] ≠ p'[-2])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> q[-1] := 0; `q'`[-1] :=  0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

0
0
a(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), ...
`a'`(n) = `+`(`*`(`p'`[-2], `*`(`^`(n, 2))), `*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `...`) (3.5.1)
 

> c(-10)=0; c(-9)=0;
 

 

0 = 0
0 = 0 (3.5.2)
 

Initial coefficient tau[-mu] 

> mu := 4; k := -8:
c(k)=0; tau[-mu] = solve( %, tau[-mu] );
 

 

 

4
`+`(`*`(`q'`[0], `*`(`^`(tau[-4], 2))), `*`(p[-2], `*`(tau[-4])), `-`(`*`(`p'`[-2], `*`(tau[-4])))) = 0
tau[-4] = (0, `+`(`-`(`/`(`*`(`+`(p[-2], `-`(`p'`[-2]))), `*`(`q'`[0]))))) (3.5.1.1)
 

> alpha = coeff( c(k) , tau[-mu], 2 );
beta  = coeff( c(k) , tau[-mu], 1 );
gamma = coeff( c(k) , tau[-mu], 0 );
 

Quadratic equation satisfied by the coefficient Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

 

 

alpha = `q'`[0]
beta = `+`(p[-2], `-`(`p'`[-2]))
gamma = 0 (3.5.1.2)
 

The next coefficients 

Choice of the first solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[1];
eq := [%]: for j from 1 to 9 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); if j<=7 or j mod 2 = 1 then print(%): end: eq := [ op(eq), % ]: end:
 

 

 

 

 

 

 

 

 

tau[-4] = 0
tau[-3] = 0
tau[-2] = 0
tau[-1] = 0
tau[0] = `/`(`*`(`p'`[-2], `*`(q[0])), `*`(`+`(p[-2], `-`(`p'`[-2]))))
tau[1] = 0
tau[2] = `+`(`-`(`/`(`*`(`+`(`*`(2, `*`(p[-2], `*`(`p'`[-2], `*`(q[0])))), `-`(`*`(`p'`[-1], `*`(q[0], `*`(p[-2])))), `-`(`*`(`p'`[-2], `*`(q[1], `*`(p[-2])))), `*`(`^`(`p'`[-2], 2), `*`(q[1])), `*`(p...
tau[3] = 0
tau[5] = 0 (3.5.2.1.1)
 

Choice of the second solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[2];
eq := [%]: for j from 1 to 9 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); if j<=5 or j mod 2 = 1 then print(%): end: eq := [ op(eq), % ]: end:
 

 

 

 

 

 

 

 

tau[-4] = `+`(`-`(`/`(`*`(`+`(p[-2], `-`(`p'`[-2]))), `*`(`q'`[0]))))
tau[-3] = 0
tau[-2] = `/`(`*`(`+`(`*`(`p'`[-1], `*`(`q'`[0])), `*`(`q'`[1], `*`(p[-2])), `-`(`*`(`q'`[1], `*`(`p'`[-2]))), `-`(`*`(p[-1], `*`(`q'`[0]))))), `*`(`^`(`q'`[0], 2)))
tau[-1] = 0
tau[0] = `/`(`*`(`+`(`*`(p[-2], `*`(`p'`[-1], `*`(`^`(`q'`[0], 2)))), `*`(`q'`[0], `*`(`q'`[1], `*`(`^`(p[-2], 2)))), `-`(`*`(`p'`[-2], `*`(p[-1], `*`(`^`(`q'`[0], 2))))), `*`(`q'`[2], `*`(`q'`[0], `*...
tau[0] = `/`(`*`(`+`(`*`(p[-2], `*`(`p'`[-1], `*`(`^`(`q'`[0], 2)))), `*`(`q'`[0], `*`(`q'`[1], `*`(`^`(p[-2], 2)))), `-`(`*`(`p'`[-2], `*`(p[-1], `*`(`^`(`q'`[0], 2))))), `*`(`q'`[2], `*`(`q'`[0], `*...
tau[0] = `/`(`*`(`+`(`*`(p[-2], `*`(`p'`[-1], `*`(`^`(`q'`[0], 2)))), `*`(`q'`[0], `*`(`q'`[1], `*`(`^`(p[-2], 2)))), `-`(`*`(`p'`[-2], `*`(p[-1], `*`(`^`(`q'`[0], 2))))), `*`(`q'`[2], `*`(`q'`[0], `*...
tau[0] = `/`(`*`(`+`(`*`(p[-2], `*`(`p'`[-1], `*`(`^`(`q'`[0], 2)))), `*`(`q'`[0], `*`(`q'`[1], `*`(`^`(p[-2], 2)))), `-`(`*`(`p'`[-2], `*`(p[-1], `*`(`^`(`q'`[0], 2))))), `*`(`q'`[2], `*`(`q'`[0], `*...
tau[0] = `/`(`*`(`+`(`*`(p[-2], `*`(`p'`[-1], `*`(`^`(`q'`[0], 2)))), `*`(`q'`[0], `*`(`q'`[1], `*`(`^`(p[-2], 2)))), `-`(`*`(`p'`[-2], `*`(p[-1], `*`(`^`(`q'`[0], 2))))), `*`(`q'`[2], `*`(`q'`[0], `*...
tau[1] = 0
tau[3] = 0
tau[5] = 0 (3.5.2.2.1)
 

Class D_{21} 

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

a(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), ...
`a'`(n) = `+`(`*`(`p'`[-2], `*`(`^`(n, 2))), `*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(...
b(n) = `+`(`*`(q[-1], `*`(n)), q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`*`(`q'`[-1], `*`(n)), `q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(... (3.6.1)
 

> c(-10)=0; tau[-4] := 0;
 

 

`*`(`q'`[-1], `*`(`^`(tau[-4], 2))) = 0
0 (3.6.2)
 

> c(-9)=0;
 

0 = 0 (3.6.3)
 

> c(-8)=0; tau[-3] := 0;
 

 

`*`(`q'`[-1], `*`(`^`(tau[-3], 2))) = 0
0 (3.6.4)
 

> c(-7)=0;
 

0 = 0 (3.6.5)
 

Initial coefficient tau[-mu] 

> mu := 2; k := -6:
c(k); tau[-mu] = solve( %, tau[-mu] );
 

 

 

2
`+`(`-`(`*`(`p'`[-2], `*`(tau[-2]))), `*`(`q'`[-1], `*`(q[-1], `*`(tau[-2]))), `*`(`q'`[-1], `*`(`^`(tau[-2], 2))), `-`(`*`(`p'`[-2], `*`(q[-1]))), `*`(p[-2], `*`(tau[-2])))
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-2], `-`(`*`(q[-1], `*`(`q'`[-1]))), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-2], `-`(`*`(q[-1], `*`(`q'`[-1]))), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-2], `-`(`*`(q[-1], `*`(`q'`[-1]))), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-2], `-`(`*`(q[-1], `*`(`q'`[-1]))), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2...
(3.6.1.1)
 

> alpha = coeff( c(k) , tau[-mu], 2 );
beta  = coeff( c(k) , tau[-mu], 1 );
gamma = coeff( c(k) , tau[-mu], 0 );
 

Quadratic equation satisfied by the coefficient Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

 

 

alpha = `q'`[-1]
beta = `+`(`-`(`p'`[-2]), `*`(q[-1], `*`(`q'`[-1])), p[-2])
gamma = `+`(`-`(`*`(`p'`[-2], `*`(q[-1])))) (3.6.1.2)
 

The next coefficients 

Choice of the first solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[1];
eq := [%]: for j from 1 to 7 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); if j<=3 or j mod 2 = 1 then print(%): end: eq := [ op(eq), % ]: end:
 

 

 

 

 

 

tau[-2] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-2], `-`(`*`(q[-1], `*`(`q'`[-1]))), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2]...
tau[-2] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-2], `-`(`*`(q[-1], `*`(`q'`[-1]))), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2]...
tau[-2] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-2], `-`(`*`(q[-1], `*`(`q'`[-1]))), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2]...
tau[-1] = 0
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[-1], `*`(`^`(p[-2], 2)))), `*`(`q'`[0], `*`(`^`(`p'`[-2], 2))), `*`(`q'`[0], `*`(`^`(p[-2], 2))), `*`(`q'`[0], `*`(q[-1], `*`(`q'`[-1], `*`(`p'...
tau[1] = 0
tau[3] = 0
tau[5] = 0 (3.6.2.1.1)
 

Choice of the second solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[2];
 

> eq := [%]: for j from 1 to 7 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); if j<=3 or j mod 2 = 1 then print(%): end: eq := [ op(eq), % ]: end:
 

 

 

 

 

 

tau[-2] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`p'`[-2]), `*`(q[-1], `*`(`q'`[-1])), p[-2], `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2],...
tau[-2] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`p'`[-2]), `*`(q[-1], `*`(`q'`[-1])), p[-2], `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2],...
tau[-2] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`p'`[-2]), `*`(q[-1], `*`(`q'`[-1])), p[-2], `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2],...
tau[-1] = 0
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[0] = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`(`q'`[0], `*`(`^`(`p'`[-2], 2)))), `-`(`*`(`q'`[0], `*`(`^`(p[-2], 2)))), `*`(`q'`[-1], `*`(`^`(p[-2], 2))), `*`(`p'`[-1], `*`(q[-1], `*`(`^`(`q'`[-1...
tau[1] = 0
tau[3] = 0
tau[5] = 0 (3.6.2.2.1)
 

The product I_n (Lemma 3.1) 

> I_n := 'a(n+1)*X(n)/(b(n+1)+X(n+1))';
 

`/`(`*`(a(`+`(n, 1)), `*`(X(n))), `*`(`+`(b(`+`(n, 1)), X(`+`(n, 1))))) (4.1)
 

Class D_{10} (p[-1] = p'[-1])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := 0; p[-2] := 0; `p'`[-1] := p[-1]; q[-1] := 0; `q'`[-1] := 0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

 

 

 

0
0
p[-1]
0
0
a(n) = `+`(`*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), `*`(`^`(n, 5))), `...`)
`a'`(n) = `+`(`*`(p[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, 4))), `/`(`*`(`p'`[5]), `*`(`^`...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `...`) (4.1.1)
 

> tau[-4] := 0;
 

> tau[-3] := 0;
 

> tau[-2] := 0;
 

 

 

0
0
0 (4.1.2)
 

Initial coefficient tau[-mu] 

> c(-2)=0; tau[-1] = solve( %, tau[-1] );
 

 

`+`(`*`(`q'`[0], `*`(`^`(tau[-1], 2))), `-`(`*`(p[-1], `*`(q[0])))) = 0
tau[-1] = (`/`(`*`(`^`(`*`(`q'`[0], `*`(p[-1], `*`(q[0]))), `/`(1, 2))), `*`(`q'`[0])), `+`(`-`(`/`(`*`(`^`(`*`(`q'`[0], `*`(p[-1], `*`(q[0]))), `/`(1, 2))), `*`(`q'`[0]))))) (4.1.1.1)
 

> 'I[n]'= asympt(I_n,n,1);
 

I[n] = `+`(`*`(p[-1], `*`(n)), `/`(`*`(`+`(`*`(p[-1], `*`(tau[0])), `-`(`*`(p[-1], `*`(`+`(q[0], tau[0])))))), `*`(tau[-1], `*`(`^`(`/`(1, `*`(n)), `/`(1, 2))))), `/`(`*`(`+`(`*`(p[-1], `*`(tau[1])), ...
I[n] = `+`(`*`(p[-1], `*`(n)), `/`(`*`(`+`(`*`(p[-1], `*`(tau[0])), `-`(`*`(p[-1], `*`(`+`(q[0], tau[0])))))), `*`(tau[-1], `*`(`^`(`/`(1, `*`(n)), `/`(1, 2))))), `/`(`*`(`+`(`*`(p[-1], `*`(tau[1])), ...
(4.1.3)
 

Observe that coefficient of n^(1/2) is 

Typesetting:-mrow(Typesetting:-mo( 

Class D_{10} (p[-1] ≠ p'[-1])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := 0; p[-2] := 0; q[-1] := 0; `q'`[-1] :=  0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

 

 

0
0
0
0
a(n) = `+`(`*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), `*`(`^`(n, 5))), `/`(`*`(p[6...
`a'`(n) = `+`(`*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, 4))), `/`(`*`(`p'`[5]), `*`(...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `/`(`*`(q[6]), `*`(`^`(n, 6))),...
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `/`(`*`(`q... (4.2.1)
 

> tau[-4] := 0;
 

> tau[-3] := 0;
 

 

0
0 (4.2.2)
 

Initial coefficient tau[-mu] 

> mu := 2; k := -4;
c(k); tau[-mu] = solve( %, tau[-mu] );
 

 

 

 

2
-4
`+`(`-`(`*`(`p'`[-1], `*`(tau[-2]))), `*`(`q'`[0], `*`(`^`(tau[-2], 2))), `*`(p[-1], `*`(tau[-2])))
tau[-2] = (0, `+`(`-`(`/`(`*`(`+`(`-`(`p'`[-1]), p[-1])), `*`(`q'`[0]))))) (4.2.1.1)
 

Choice of the first solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[1]:
eq := [%]: for j from 1 to 4 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); eq := [ op(eq), % ]: end: eq;
 

[tau[-2] = 0, tau[-1] = 0, tau[0] = `/`(`*`(`p'`[-1], `*`(q[0])), `*`(`+`(`-`(`p'`[-1]), p[-1]))), tau[1] = 0, tau[2] = `+`(`-`(`/`(`*`(`+`(`*`(`q'`[0], `*`(`^`(q[0], 2), `*`(`p'`[-1], `*`(p[-1])))), ...
[tau[-2] = 0, tau[-1] = 0, tau[0] = `/`(`*`(`p'`[-1], `*`(q[0])), `*`(`+`(`-`(`p'`[-1]), p[-1]))), tau[1] = 0, tau[2] = `+`(`-`(`/`(`*`(`+`(`*`(`q'`[0], `*`(`^`(q[0], 2), `*`(`p'`[-1], `*`(p[-1])))), ...
[tau[-2] = 0, tau[-1] = 0, tau[0] = `/`(`*`(`p'`[-1], `*`(q[0])), `*`(`+`(`-`(`p'`[-1]), p[-1]))), tau[1] = 0, tau[2] = `+`(`-`(`/`(`*`(`+`(`*`(`q'`[0], `*`(`^`(q[0], 2), `*`(`p'`[-1], `*`(p[-1])))), ...
(4.2.2.1)
 

> 'I[n]'=applyop(simplify, {1,2}, asympt(subs({op(eq)}, I_n),n,2));
 

I[n] = `+`(`*`(`p'`[-1], `*`(n)), `/`(`*`(`+`(`*`(`p'`[0], `*`(p[-1])), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`q'`[0])))), `-`(`*`(`p'`[0], `*`(`p'`[-1]))))), `*`(`+`(`-`(`p'`[-1]), p[-1]))), `/`(`*`(`+`(`*...
I[n] = `+`(`*`(`p'`[-1], `*`(n)), `/`(`*`(`+`(`*`(`p'`[0], `*`(p[-1])), `-`(`*`(`p'`[-1], `*`(q[0], `*`(`q'`[0])))), `-`(`*`(`p'`[0], `*`(`p'`[-1]))))), `*`(`+`(`-`(`p'`[-1]), p[-1]))), `/`(`*`(`+`(`*...
(4.2.2.2)
 

Choice of the second solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[2]:
eq := [%]: for j from 1 to 1 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); eq := [ op(eq), % ]: end: eq;
 

[tau[-2] = `+`(`-`(`/`(`*`(`+`(`-`(`p'`[-1]), p[-1])), `*`(`q'`[0])))), tau[-1] = 0] (4.2.3.1)
 

> 'I[n]'=applyop(simplify, {1,2}, asympt(subs({op(eq)}, I_n),n,1));
 

I[n] = `+`(`*`(p[-1], `*`(n)), `/`(`*`(`+`(`-`(`*`(p[0], `*`(`p'`[-1]))), `*`(p[0], `*`(p[-1])), `*`(p[-1], `*`(q[0], `*`(`q'`[0]))))), `*`(`+`(`-`(`p'`[-1]), p[-1]))), O(`/`(1, `*`(n)))) (4.2.3.2)
 

Class D_{11} 

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := 0; p[-2] := 0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

0
0
a(n) = `+`(`*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), `*`(`^`(n, 5))), `...`)
`a'`(n) = `+`(`*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, 4))), `/`(`*`(`p'`[5]), `*`(...
b(n) = `+`(`*`(q[-1], `*`(n)), q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`*`(`q'`[-1], `*`(n)), `q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(... (4.3.1)
 

> tau[-4] := 0;
 

> tau[-3] := 0;
 

 

0
0 (4.3.2)
 

Initial coefficient tau[-mu] 

> mu := 2; k := -6:
c(k); tau[-mu] = solve( %, tau[-mu] );
 

 

 

2
`+`(`*`(`q'`[-1], `*`(q[-1], `*`(tau[-2]))), `*`(`q'`[-1], `*`(`^`(tau[-2], 2))))
tau[-2] = (0, `+`(`-`(q[-1]))) (4.3.1.1)
 

Choice of the first solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[1]:
eq := [%]: for j from 1 to 3 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); eq := [ op(eq), % ]: end: eq;
 

[tau[-2] = 0, tau[-1] = 0, tau[0] = `/`(`*`(`p'`[-1]), `*`(`q'`[-1])), tau[1] = 0] (4.3.2.1)
 

> 'I[n]'=asympt(subs({op(eq)}, I_n),n,1);
 

I[n] = `+`(`/`(`*`(p[-1], `*`(`p'`[-1])), `*`(`q'`[-1], `*`(q[-1]))), O(`/`(1, `*`(n)))) (4.3.2.2)
 

Choice of the second solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[2]:
eq := [%]: for j from 1 to 3 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); eq := [ op(eq), % ]: end: eq;
 

[tau[-2] = `+`(`-`(q[-1])), tau[-1] = 0, tau[0] = `+`(`-`(`/`(`*`(`+`(p[-1], `*`(`q'`[-1], `*`(q[0])))), `*`(`q'`[-1])))), tau[1] = 0] (4.3.3.1)
 

> 'I[n]'= simplify( asympt(subs({op(eq)}, I_n),n,1) );
 

I[n] = `+`(`*`(q[-1], `*`(`q'`[-1], `*`(`^`(n, 2)))), O(n)) (4.3.3.2)
 

Class D_{20} (p[-2] = p'[-2])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := p[-2]; q[-1] := 0; `q'`[-1] :=  0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

 

p[-2]
0
0
a(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), ...
`a'`(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, ...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `...`) (4.4.1)
 

> tau[-4] := 0;
 

> tau[-3] := 0;
 

 

0
0 (4.4.2)
 

Initial coefficient tau[-mu] 

> mu := 2; k := -4:
c(k)=0; tau[-mu] = solve( %, tau[-mu] );
 

 

 

2
`+`(`-`(`*`(`p'`[-1], `*`(tau[-2]))), `*`(p[-1], `*`(tau[-2])), `*`(`q'`[0], `*`(`^`(tau[-2], 2))), `-`(`*`(p[-2], `*`(q[0]))), `*`(p[-2], `*`(tau[-2]))) = 0
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-1], `-`(p[-1]), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-1], `-`(p[-1]), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-1], `-`(p[-1]), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-1], `-`(p[-1]), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-1], 2)), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-1])))), `-`(`*`(2, `*`(`p'`[-1], `*`(p[-2])))), `*`(`^`(p...
(4.4.1.1)
 

> 'I[n]'=collect( convert(asympt(subs({tau[-1]=0},I_n),n,2),polynom), n, simplify ) + `...`;
 

I[n] = `+`(`*`(p[-2], `*`(`^`(n, 2))), `-`(`/`(`*`(`+`(`-`(`*`(p[-2], `*`(tau[-2]))), `-`(`*`(p[-1], `*`(tau[-2]))), `*`(p[-2], `*`(q[0]))), `*`(n)), `*`(tau[-2]))), `/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`...
I[n] = `+`(`*`(p[-2], `*`(`^`(n, 2))), `-`(`/`(`*`(`+`(`-`(`*`(p[-2], `*`(tau[-2]))), `-`(`*`(p[-1], `*`(tau[-2]))), `*`(p[-2], `*`(q[0]))), `*`(n)), `*`(tau[-2]))), `/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`...
I[n] = `+`(`*`(p[-2], `*`(`^`(n, 2))), `-`(`/`(`*`(`+`(`-`(`*`(p[-2], `*`(tau[-2]))), `-`(`*`(p[-1], `*`(tau[-2]))), `*`(p[-2], `*`(q[0]))), `*`(n)), `*`(tau[-2]))), `/`(`*`(`/`(1, 2), `*`(`+`(`-`(`*`...
(4.4.3)
 

Class D_{20} (p[-2] ≠ p'[-2])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> q[-1] := 0; `q'`[-1] :=  0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

0
0
a(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), ...
`a'`(n) = `+`(`*`(`p'`[-2], `*`(`^`(n, 2))), `*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `...`) (4.5.1)
 

Initial coefficient tau[-mu] 

> mu := 4; k := -8:
c(k)=0; tau[-mu] = solve( %, tau[-mu] );
 

 

 

4
`+`(`*`(`q'`[0], `*`(`^`(tau[-4], 2))), `*`(p[-2], `*`(tau[-4])), `-`(`*`(`p'`[-2], `*`(tau[-4])))) = 0
tau[-4] = (0, `+`(`-`(`/`(`*`(`+`(p[-2], `-`(`p'`[-2]))), `*`(`q'`[0]))))) (4.5.1.1)
 

Choice of the first solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[1]:
eq := [%]: for j from 1 to 5 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); eq := [ op(eq), % ]: end: eq;
 

[tau[-4] = 0, tau[-3] = 0, tau[-2] = 0, tau[-1] = 0, tau[0] = `/`(`*`(`p'`[-2], `*`(q[0])), `*`(`+`(p[-2], `-`(`p'`[-2])))), tau[1] = 0] (4.5.2.1)
 

> 'I[n]'=simplify(asympt(subs({op(eq)}, I_n),n,1));
 

I[n] = `+`(`*`(`p'`[-2], `*`(`^`(n, 2))), O(n)) (4.5.2.2)
 

Choice of the second solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[2]:
eq := [%]: for j from 1 to 2 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); eq := [ op(eq), % ]: end: eq;
 

[tau[-4] = `+`(`-`(`/`(`*`(`+`(p[-2], `-`(`p'`[-2]))), `*`(`q'`[0])))), tau[-3] = 0, tau[-2] = `/`(`*`(`+`(`*`(`p'`[-1], `*`(`q'`[0])), `*`(`q'`[1], `*`(p[-2])), `-`(`*`(`q'`[1], `*`(`p'`[-2]))), `-`(... (4.5.3.1)
 

> 'I[n]'=asympt(subs({op([])}, I_n),n,1);
 

I[n] = `+`(`*`(p[-2], `*`(`^`(n, 2))), `/`(`*`(`+`(`*`(p[-2], `*`(tau[-2])), `*`(`+`(`*`(2, `*`(p[-2])), p[-1]), `*`(tau[-4])), `-`(`*`(p[-2], `*`(`+`(tau[-2], `*`(2, `*`(tau[-4]))))))), `*`(n)), `*`(...
I[n] = `+`(`*`(p[-2], `*`(`^`(n, 2))), `/`(`*`(`+`(`*`(p[-2], `*`(tau[-2])), `*`(`+`(`*`(2, `*`(p[-2])), p[-1]), `*`(tau[-4])), `-`(`*`(p[-2], `*`(`+`(tau[-2], `*`(2, `*`(tau[-4]))))))), `*`(n)), `*`(...
I[n] = `+`(`*`(p[-2], `*`(`^`(n, 2))), `/`(`*`(`+`(`*`(p[-2], `*`(tau[-2])), `*`(`+`(`*`(2, `*`(p[-2])), p[-1]), `*`(tau[-4])), `-`(`*`(p[-2], `*`(`+`(tau[-2], `*`(2, `*`(tau[-4]))))))), `*`(n)), `*`(...
I[n] = `+`(`*`(p[-2], `*`(`^`(n, 2))), `/`(`*`(`+`(`*`(p[-2], `*`(tau[-2])), `*`(`+`(`*`(2, `*`(p[-2])), p[-1]), `*`(tau[-4])), `-`(`*`(p[-2], `*`(`+`(tau[-2], `*`(2, `*`(tau[-4]))))))), `*`(n)), `*`(...
I[n] = `+`(`*`(p[-2], `*`(`^`(n, 2))), `/`(`*`(`+`(`*`(p[-2], `*`(tau[-2])), `*`(`+`(`*`(2, `*`(p[-2])), p[-1]), `*`(tau[-4])), `-`(`*`(p[-2], `*`(`+`(tau[-2], `*`(2, `*`(tau[-4]))))))), `*`(n)), `*`(...
I[n] = `+`(`*`(p[-2], `*`(`^`(n, 2))), `/`(`*`(`+`(`*`(p[-2], `*`(tau[-2])), `*`(`+`(`*`(2, `*`(p[-2])), p[-1]), `*`(tau[-4])), `-`(`*`(p[-2], `*`(`+`(tau[-2], `*`(2, `*`(tau[-4]))))))), `*`(n)), `*`(...
I[n] = `+`(`*`(p[-2], `*`(`^`(n, 2))), `/`(`*`(`+`(`*`(p[-2], `*`(tau[-2])), `*`(`+`(`*`(2, `*`(p[-2])), p[-1]), `*`(tau[-4])), `-`(`*`(p[-2], `*`(`+`(tau[-2], `*`(2, `*`(tau[-4]))))))), `*`(n)), `*`(...
I[n] = `+`(`*`(p[-2], `*`(`^`(n, 2))), `/`(`*`(`+`(`*`(p[-2], `*`(tau[-2])), `*`(`+`(`*`(2, `*`(p[-2])), p[-1]), `*`(tau[-4])), `-`(`*`(p[-2], `*`(`+`(tau[-2], `*`(2, `*`(tau[-4]))))))), `*`(n)), `*`(...
(4.5.3.2)
 

Class D_{21} 

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

a(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), ...
`a'`(n) = `+`(`*`(`p'`[-2], `*`(`^`(n, 2))), `*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(...
b(n) = `+`(`*`(q[-1], `*`(n)), q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`*`(`q'`[-1], `*`(n)), `q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(... (4.6.1)
 

> tau[-4] := 0;
 

> tau[-3] := 0;
 

 

0
0 (4.6.2)
 

Initial coefficient tau[-mu] 

> mu := 2; k := -6:
c(k); tau[-mu] = solve( %, tau[-mu] );
 

 

 

2
`+`(`-`(`*`(`p'`[-2], `*`(tau[-2]))), `*`(`q'`[-1], `*`(q[-1], `*`(tau[-2]))), `*`(`q'`[-1], `*`(`^`(tau[-2], 2))), `-`(`*`(`p'`[-2], `*`(q[-1]))), `*`(p[-2], `*`(tau[-2])))
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-2], `-`(`*`(q[-1], `*`(`q'`[-1]))), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-2], `-`(`*`(q[-1], `*`(`q'`[-1]))), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-2], `-`(`*`(q[-1], `*`(`q'`[-1]))), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2...
tau[-2] = (`+`(`/`(`*`(`/`(1, 2), `*`(`+`(`p'`[-2], `-`(`*`(q[-1], `*`(`q'`[-1]))), `-`(p[-2]), `*`(`^`(`+`(`*`(`^`(`p'`[-2], 2)), `*`(2, `*`(`q'`[-1], `*`(`p'`[-2], `*`(q[-1])))), `-`(`*`(2, `*`(p[-2...
(4.6.1.1)
 

> 'I[n]'=collect( convert(asympt(subs({tau[-1]=0},I_n),n,1),polynom), n, simplify ) + `...`;
 

I[n] = `+`(`/`(`*`(p[-2], `*`(tau[-2], `*`(`^`(n, 2)))), `*`(`+`(q[-1], tau[-2]))), `/`(`*`(`+`(`*`(p[-2], `*`(tau[0], `*`(q[-1]))), `*`(tau[-2], `*`(p[-2], `*`(q[-1]))), `*`(`^`(tau[-2], 2), `*`(p[-2...
I[n] = `+`(`/`(`*`(p[-2], `*`(tau[-2], `*`(`^`(n, 2)))), `*`(`+`(q[-1], tau[-2]))), `/`(`*`(`+`(`*`(p[-2], `*`(tau[0], `*`(q[-1]))), `*`(tau[-2], `*`(p[-2], `*`(q[-1]))), `*`(`^`(tau[-2], 2), `*`(p[-2...
I[n] = `+`(`/`(`*`(p[-2], `*`(tau[-2], `*`(`^`(n, 2)))), `*`(`+`(q[-1], tau[-2]))), `/`(`*`(`+`(`*`(p[-2], `*`(tau[0], `*`(q[-1]))), `*`(tau[-2], `*`(p[-2], `*`(q[-1]))), `*`(`^`(tau[-2], 2), `*`(p[-2...
(4.6.3)
 

>
 

Coefficients Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( in the proof of Theorem 4.1 

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `u'` := unapply( X(n), n );
 

proc (n) options operator, arrow; `+`(`*`(tau[-4], `*`(`^`(n, 2))), `*`(tau[-3], `*`(`^`(n, `/`(3, 2)))), `*`(tau[-2], `*`(n)), `*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n... (5.1)
 

> `phi'`[n] := 1 + m/(2*n);
`psi'`[n] := ' `a'`(n)*a(n+1) / (a(n+1)+`b'`(n)*b(n+1) + `b'`(n)*`u'`(n+1))^2 ';
 

 

`+`(1, `/`(`*`(`/`(1, 2), `*`(m)), `*`(n)))
`/`(`*`(`a'`(n), `*`(a(`+`(n, 1)))), `*`(`^`(`+`(a(`+`(n, 1)), `*`(`b'`(n), `*`(b(`+`(n, 1)))), `*`(`b'`(n), `*`(`u'`(`+`(n, 1))))), 2))) (5.2)
 

Class D_{10} (p[-1] = p'[-1])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := 0; p[-2] := 0; `p'`[-1] := p[-1]; q[-1] := 0; `q'`[-1] := 0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

 

 

 

0
0
p[-1]
0
0
a(n) = `+`(`*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), `*`(`^`(n, 5))), `...`)
`a'`(n) = `+`(`*`(p[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, 4))), `/`(`*`(`p'`[5]), `*`(`^`...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `...`) (5.1.1)
 

> tau[-4] := 0;
 

> tau[-3] := 0;
 

> tau[-2] := 0;
 

 

 

0
0
0 (5.1.2)
 

> '`u'`'(n) = `u'`(n)+`...`;
 

`u'`(n) = `+`(`*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n, `/`(1, 2)))), `/`(`*`(tau[2]), `*`(n)), `/`(`*`(tau[3]), `*`(`^`(n, `/`(3, 2)))), `/`(`*`(tau[4]), `*`(`^`(n, 2)... (5.1.3)
 

> '`phi'`[n] - `psi'`[n]' = asympt( `phi'`[n] - `psi'`[n] , n, 3 );
 

`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(`/`(`*`(2, `*`(`q'`[0], `*`(tau[-1], `*`(`^`(`/`(1, `*`(n)), `/`(1, 2)))))), `*`(p[-1])), O(`/`(1, `*`(n)))) (5.1.4)
 

Class D_{10} (p[-1] ≠ p'[-1])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := 0; p[-2] := 0; q[-1] := 0; `q'`[-1] :=  0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

 

 

0
0
0
0
a(n) = `+`(`*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), `*`(`^`(n, 5))), `...`)
`a'`(n) = `+`(`*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, 4))), `/`(`*`(`p'`[5]), `*`(...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `...`) (5.2.1)
 

> tau[-4] := 0;
 

> tau[-3] := 0;
 

 

0
0 (5.2.2)
 

> '`u'`'(n) = `u'`(n)+`...`;
 

`u'`(n) = `+`(`*`(tau[-2], `*`(n)), `*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n, `/`(1, 2)))), `/`(`*`(tau[2]), `*`(n)), `/`(`*`(tau[3]), `*`(`^`(n, `/`(3, 2)))), `/`(`*`(... (5.2.3)
 

> '`phi'`[n] - `psi'`[n]' = asympt( `phi'`[n] - `psi'`[n] , n, 3 );
 

`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(1, `-`(`/`(`*`(`p'`[-1], `*`(p[-1])), `*`(`^`(`+`(p[-1], `*`(`q'`[0], `*`(tau[-2]))), 2)))), `/`(`*`(2, `*`(`p'`[-1], `*`(p[-1], `*`(`q'`[0], `*`(tau[-1], `*`(`^`(... (5.2.4)
 

Initial coefficient tau[-mu] 

> mu := 2; k := -4;
c(k); tau[-mu] = solve( %, tau[-mu] );
 

 

 

 

2
-4
`+`(`-`(`*`(`p'`[-1], `*`(tau[-2]))), `*`(p[-1], `*`(tau[-2])), `*`(`q'`[0], `*`(`^`(tau[-2], 2))))
tau[-2] = (0, `+`(`-`(`/`(`*`(`+`(`-`(`p'`[-1]), p[-1])), `*`(`q'`[0]))))) (5.2.1.1)
 

Choice of the first solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[1]:
eq := [%]: for j from 1 to 3 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); eq := [ op(eq), % ]: end: eq;
 

[tau[-2] = 0, tau[-1] = 0, tau[0] = `/`(`*`(`p'`[-1], `*`(q[0])), `*`(`+`(`-`(`p'`[-1]), p[-1]))), tau[1] = 0] (5.2.2.1)
 

> '`phi'`[n] - `psi'`[n]'=asympt(subs({op(eq)}, `phi'`[n] - `psi'`[n]),n,3);
 

`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(1, `-`(`/`(`*`(`p'`[-1]), `*`(p[-1]))), O(`/`(1, `*`(n)))) (5.2.2.2)
 

Choice of the second solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[2]:
eq := [%]: for j from 1 to 3 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); eq := [ op(eq), % ]: end: eq;
 

[tau[-2] = `+`(`-`(`/`(`*`(`+`(`-`(`p'`[-1]), p[-1])), `*`(`q'`[0])))), tau[-1] = 0, tau[0] = `/`(`*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `-`(`*`(2, `*`(`q'`[1], `*`(`p'`[-1], `*`(p[-1]))))), `*`...
[tau[-2] = `+`(`-`(`/`(`*`(`+`(`-`(`p'`[-1]), p[-1])), `*`(`q'`[0])))), tau[-1] = 0, tau[0] = `/`(`*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `-`(`*`(2, `*`(`q'`[1], `*`(`p'`[-1], `*`(p[-1]))))), `*`...
[tau[-2] = `+`(`-`(`/`(`*`(`+`(`-`(`p'`[-1]), p[-1])), `*`(`q'`[0])))), tau[-1] = 0, tau[0] = `/`(`*`(`+`(`*`(`q'`[1], `*`(`^`(`p'`[-1], 2))), `-`(`*`(2, `*`(`q'`[1], `*`(`p'`[-1], `*`(p[-1]))))), `*`...
(5.2.3.1)
 

> '`phi'`[n] - `psi'`[n]'=asympt(subs({op(eq)}, `phi'`[n] - `psi'`[n]),n,3);
 

`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(1, `-`(`/`(`*`(p[-1]), `*`(`p'`[-1]))), O(`/`(1, `*`(n)))) (5.2.3.2)
 

Class D_{11} 

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := 0; p[-2] := 0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

0
0
a(n) = `+`(`*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), `*`(`^`(n, 5))), `...`)
`a'`(n) = `+`(`*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, 4))), `/`(`*`(`p'`[5]), `*`(...
b(n) = `+`(`*`(q[-1], `*`(n)), q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`*`(`q'`[-1], `*`(n)), `q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(... (5.3.1)
 

> tau[-4] := 0;
 

> tau[-3] := 0;
 

 

0
0 (5.3.2)
 

> '`u'`'(n) = `u'`(n)+`...`;
 

`u'`(n) = `+`(`*`(tau[-2], `*`(n)), `*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n, `/`(1, 2)))), `/`(`*`(tau[2]), `*`(n)), `/`(`*`(tau[3]), `*`(`^`(n, `/`(3, 2)))), `/`(`*`(... (5.3.3)
 

> '`phi'`[n] - `psi'`[n]' = asympt( `phi'`[n] - `psi'`[n] , n, 1 );
 

`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(1, O(`/`(1, `*`(n)))) (5.3.4)
 

Class D_{20} (p[-2] = p'[-2])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> `p'`[-2] := p[-2]; q[-1] := 0; `q'`[-1] :=  0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

 

p[-2]
0
0
a(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), ...
`a'`(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(n, ...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `...`) (5.4.1)
 

> tau[-4] := 0;
 

> tau[-3] := 0;
 

 

0
0 (5.4.2)
 

> '`u'`'(n) = `u'`(n)+`...`;
 

`u'`(n) = `+`(`*`(tau[-2], `*`(n)), `*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n, `/`(1, 2)))), `/`(`*`(tau[2]), `*`(n)), `/`(`*`(tau[3]), `*`(`^`(n, `/`(3, 2)))), `/`(`*`(... (5.4.3)
 

> '`phi'`[n] - `psi'`[n]' = asympt( `phi'`[n] - `psi'`[n] , n, 6 );
 

`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(`/`(`*`(`+`(`*`(`/`(1, 2), `*`(m)), `/`(`*`(2, `*`(`+`(`*`(2, `*`(p[-2])), p[-1], `*`(`q'`[0], `*`(tau[-2]))))), `*`(p[-2])), `-`(`/`(`*`(`+`(`*`(p[-2], `*`(`+`(`*...
`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(`/`(`*`(`+`(`*`(`/`(1, 2), `*`(m)), `/`(`*`(2, `*`(`+`(`*`(2, `*`(p[-2])), p[-1], `*`(`q'`[0], `*`(tau[-2]))))), `*`(p[-2])), `-`(`/`(`*`(`+`(`*`(p[-2], `*`(`+`(`*...
(5.4.4)
 

> sigma[2] := ( op(1,rhs(%)) )*n;
 

`+`(`*`(`/`(1, 2), `*`(m)), `/`(`*`(2, `*`(`+`(`*`(2, `*`(p[-2])), p[-1], `*`(`q'`[0], `*`(tau[-2]))))), `*`(p[-2])), `-`(`/`(`*`(`+`(`*`(p[-2], `*`(`+`(`*`(2, `*`(p[-2])), p[-1]))), `*`(`p'`[-1], `*`... (5.4.5)
 

> 'sigma[2]' = simplify( sigma[2] );
 

sigma[2] = `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`*`(m, `*`(p[-2])), `*`(4, `*`(p[-2])), `*`(2, `*`(p[-1])), `*`(4, `*`(`q'`[0], `*`(tau[-2]))), `-`(`*`(2, `*`(`p'`[-1])))))), `*`(p[-2]))) (5.4.6)
 

Class D_{20} (p[-2] ≠ p'[-2])  

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> q[-1] := 0; `q'`[-1] :=  0;
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

 

 

0
0
a(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), ...
`a'`(n) = `+`(`*`(`p'`[-2], `*`(`^`(n, 2))), `*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(...
b(n) = `+`(q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(`^`(n, 5))), `...`) (5.5.1)
 

> '`u'`'(n) = `u'`(n)+`...`;
 

`u'`(n) = `+`(`*`(tau[-4], `*`(`^`(n, 2))), `*`(tau[-3], `*`(`^`(n, `/`(3, 2)))), `*`(tau[-2], `*`(n)), `*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n, `/`(1, 2)))), `/`(`*`(...
`u'`(n) = `+`(`*`(tau[-4], `*`(`^`(n, 2))), `*`(tau[-3], `*`(`^`(n, `/`(3, 2)))), `*`(tau[-2], `*`(n)), `*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n, `/`(1, 2)))), `/`(`*`(...
(5.5.2)
 

> '`phi'`[n] - `psi'`[n]' = asympt( `phi'`[n] - `psi'`[n] , n, 5 );
 

`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(1, `-`(`/`(`*`(p[-2], `*`(`p'`[-2])), `*`(`^`(`+`(p[-2], `*`(`q'`[0], `*`(tau[-4]))), 2)))), `/`(`*`(2, `*`(p[-2], `*`(`p'`[-2], `*`(`q'`[0], `*`(tau[-3], `*`(`^`(... (5.5.3)
 

Initial coefficient tau[-mu] 

> mu := 4; k := -8:
c(k)=0; tau[-mu] = solve( %, tau[-mu] );
 

 

 

4
`+`(`*`(`q'`[0], `*`(`^`(tau[-4], 2))), `*`(p[-2], `*`(tau[-4])), `-`(`*`(`p'`[-2], `*`(tau[-4])))) = 0
tau[-4] = (0, `+`(`-`(`/`(`*`(`+`(p[-2], `-`(`p'`[-2]))), `*`(`q'`[0]))))) (5.5.1.1)
 

Choice of the first solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[1]:
eq := [%]: for j from 1 to 5 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); eq := [ op(eq), % ]: end: eq;
 

[tau[-4] = 0, tau[-3] = 0, tau[-2] = 0, tau[-1] = 0, tau[0] = `/`(`*`(`p'`[-2], `*`(q[0])), `*`(`+`(p[-2], `-`(`p'`[-2])))), tau[1] = 0] (5.5.2.1)
 

> '`phi'`[n] - `psi'`[n]'=asympt(subs({op(eq)}, `phi'`[n] - `psi'`[n]),n,5);
 

`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(1, `-`(`/`(`*`(`p'`[-2]), `*`(p[-2]))), O(`/`(1, `*`(n)))) (5.5.2.2)
 

Choice of the second solution 

> tau[-mu ] = solve( c(k), tau[-mu] )[2]:
eq := [%]: for j from 1 to 2 do tau[-mu+j] = solve( subs( {op(eq)}, c(k+j) ), tau[-mu+j] ); eq := [ op(eq), % ]: end: eq;
 

[tau[-4] = `+`(`-`(`/`(`*`(`+`(p[-2], `-`(`p'`[-2]))), `*`(`q'`[0])))), tau[-3] = 0, tau[-2] = `/`(`*`(`+`(`*`(`p'`[-1], `*`(`q'`[0])), `*`(`q'`[1], `*`(p[-2])), `-`(`*`(`q'`[1], `*`(`p'`[-2]))), `-`(... (5.5.3.1)
 

> '`phi'`[n] - `psi'`[n]'=asympt(subs({op(eq)}, `phi'`[n] - `psi'`[n]),n,5);
 

`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(1, `-`(`/`(`*`(p[-2]), `*`(`p'`[-2]))), O(`/`(1, `*`(n)))) (5.5.3.2)
 

Class D_{21} 

> unassign( 'p', '`p'`', 'q', '`q'`', 'tau' );
 

> 'a'(n) = a(n)+`...`; '`a'`'(n) = `a'`(n)+`...`;
 

> 'b'(n) = b(n)+`...`; '`b'`'(n) = `b'`(n)+`...`;
 

 

 

 

a(n) = `+`(`*`(p[-2], `*`(`^`(n, 2))), `*`(p[-1], `*`(n)), p[0], `/`(`*`(p[1]), `*`(n)), `/`(`*`(p[2]), `*`(`^`(n, 2))), `/`(`*`(p[3]), `*`(`^`(n, 3))), `/`(`*`(p[4]), `*`(`^`(n, 4))), `/`(`*`(p[5]), ...
`a'`(n) = `+`(`*`(`p'`[-2], `*`(`^`(n, 2))), `*`(`p'`[-1], `*`(n)), `p'`[0], `/`(`*`(`p'`[1]), `*`(n)), `/`(`*`(`p'`[2]), `*`(`^`(n, 2))), `/`(`*`(`p'`[3]), `*`(`^`(n, 3))), `/`(`*`(`p'`[4]), `*`(`^`(...
b(n) = `+`(`*`(q[-1], `*`(n)), q[0], `/`(`*`(q[1]), `*`(n)), `/`(`*`(q[2]), `*`(`^`(n, 2))), `/`(`*`(q[3]), `*`(`^`(n, 3))), `/`(`*`(q[4]), `*`(`^`(n, 4))), `/`(`*`(q[5]), `*`(`^`(n, 5))), `...`)
`b'`(n) = `+`(`*`(`q'`[-1], `*`(n)), `q'`[0], `/`(`*`(`q'`[1]), `*`(n)), `/`(`*`(`q'`[2]), `*`(`^`(n, 2))), `/`(`*`(`q'`[3]), `*`(`^`(n, 3))), `/`(`*`(`q'`[4]), `*`(`^`(n, 4))), `/`(`*`(`q'`[5]), `*`(... (5.6.1)
 

> tau[-4] := 0;
 

> tau[-3] := 0;
 

 

0
0 (5.6.2)
 

> '`u'`'(n) = `u'`(n)+`...`;
 

`u'`(n) = `+`(`*`(tau[-2], `*`(n)), `*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n, `/`(1, 2)))), `/`(`*`(tau[2]), `*`(n)), `/`(`*`(tau[3]), `*`(`^`(n, `/`(3, 2)))), `/`(`*`(... (5.6.3)
 

> '`phi'`[n] - `psi'`[n]' = asympt( `phi'`[n] - `psi'`[n] , n, 5 );
 

`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(1, `-`(`/`(`*`(p[-2], `*`(`p'`[-2])), `*`(`^`(`+`(p[-2], `*`(`q'`[-1], `*`(tau[-2])), `*`(q[-1], `*`(`q'`[-1]))), 2)))), `/`(`*`(2, `*`(p[-2], `*`(`p'`[-2], `*`(`q...
`+`(`phi'`[n], `-`(`psi'`[n])) = `+`(1, `-`(`/`(`*`(p[-2], `*`(`p'`[-2])), `*`(`^`(`+`(p[-2], `*`(`q'`[-1], `*`(tau[-2])), `*`(q[-1], `*`(`q'`[-1]))), 2)))), `/`(`*`(2, `*`(p[-2], `*`(`p'`[-2], `*`(`q...
(5.6.4)
 

>
 

Theorem 4.1 (examples) 

Class D_{10} (p[-1] = p'[-1])  

> restart;
 

> a := n -> 3*n - 5 + 6/n;         `a'` := n -> 3*n-8+2/n+4/n^2;
b := n -> 1 + 2/n;               `b'` := n -> 8+1/n+1/n^2;
 

> p[-1] := coeff( a(n), n, 1 );    `p'`[-1] := coeff( `a'`(n), n, 1 );     if not p[-1]=`p'`[-1]    then error("Coefficients p[-1], p'[-1] should be the same."); end;
 

> q[0]  := coeff( b(n), n, 0 );    `q'`[0] := coeff( `b'`(n), n, 0 );      if q[0]*`q'`[0]/p[-1]<=0 then error("CF not in the class D_{10}^=."); end;
 

 

 

 

 

 

 

 

proc (n) options operator, arrow; `+`(`*`(3, `*`(n)), `-`(5), `/`(`*`(6), `*`(n))) end proc
proc (n) options operator, arrow; `+`(`*`(3, `*`(n)), `-`(8), `/`(`*`(2), `*`(n)), `/`(`*`(4), `*`(`^`(n, 2)))) end proc
proc (n) options operator, arrow; `+`(1, `/`(`*`(2), `*`(n))) end proc
proc (n) options operator, arrow; `+`(8, `/`(1, `*`(n)), `/`(1, `*`(`^`(n, 2)))) end proc
3
3
1
8 (6.1.1)
 

Asymptotic expansions of tails 

> EQ := '(`b'`(n)*b(n+1) + a(n+1))*u(n) + `b'`(n)*u(n)*u(n+1) - `a'`(n)*u(n+1) - `a'`(n)*b(n+1)' = 0;
 

Bilinear equation satisfied by tails of CF 

`+`(`*`(`+`(`*`(`b'`(n), `*`(b(`+`(n, 1)))), a(`+`(n, 1))), `*`(u(n))), `*`(`b'`(n), `*`(u(n), `*`(u(`+`(n, 1))))), `-`(`*`(`a'`(n), `*`(u(`+`(n, 1))))), `-`(`*`(`a'`(n), `*`(b(`+`(n, 1)))))) = 0
`+`(`*`(`+`(`*`(`b'`(n), `*`(b(`+`(n, 1)))), a(`+`(n, 1))), `*`(u(n))), `*`(`b'`(n), `*`(u(n), `*`(u(`+`(n, 1))))), `-`(`*`(`a'`(n), `*`(u(`+`(n, 1))))), `-`(`*`(`a'`(n), `*`(b(`+`(n, 1)))))) = 0
(6.1.1.1)
 

> MAX := 10:
 

> u := unapply( add( tau[j]/n^(j/2) , j=-4..MAX ), n );    # X_n
 

Asymptotic expansion of the solutions of (2.1). 

proc (n) options operator, arrow; `+`(`*`(tau[-4], `*`(`^`(n, 2))), `*`(tau[-3], `*`(`^`(n, `/`(3, 2)))), `*`(tau[-2], `*`(n)), `*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n...
proc (n) options operator, arrow; `+`(`*`(tau[-4], `*`(`^`(n, 2))), `*`(tau[-3], `*`(`^`(n, `/`(3, 2)))), `*`(tau[-2], `*`(n)), `*`(tau[-1], `*`(`^`(n, `/`(1, 2)))), tau[0], `/`(`*`(tau[1]), `*`(`^`(n...
(6.1.1.2)
 

We calculate the coefficients Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(of Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi( in the left hand side of equation (2.1). 

Observe that LHS of (2.1) is at most Typesetting:-mrow(Typesetting:-mi(, so we calculate Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(for Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

> unassign( 'tau' ):
EQ2 := simplify( subs( n=N^2, EQ ) ) assuming N>0:         # Warning: this can take some time
LHS := convert( asympt( lhs(EQ2), N, 10*MAX ), polynom ):  # Warning: this can take some time
c := m -> coeff( LHS, N, -m );
 

> tau[-4] := 0:
 

> tau[-3] := 0:
 

> tau[-2] := 0:
 

> tau[-1] := signum(Re(`q'`[0]/p[-1]*sqrt(q[0]*p[-1]/`q'`[0]))) *  sqrt( q[0]*p[-1]/`q'`[0] );
 

 

proc (m) options operator, arrow; coeff(LHS, N, `+`(`-`(m))) end proc
`+`(`*`(`/`(1, 4), `*`(`^`(6, `/`(1, 2))))) (6.1.1.3)
 

1 (6.1.1.4)
 

> mu := 1: k := -2:
 

> for j from 1 to MAX+mu-3 do tau[-mu+j] := solve( c(k+j), tau[-mu+j] ); end:
 

> u := unapply( add( tau[j]/n^(j/2), j=-1..10 ), n );
 

proc (n) options operator, arrow; `+`(`-`(`/`(25, 32)), `*`(`/`(1, 4), `*`(`^`(6, `/`(1, 2)), `*`(`^`(n, `/`(1, 2))))), `/`(`*`(`/`(93, 1024), `*`(`^`(6, `/`(1, 2)))), `*`(`^`(n, `/`(1, 2)))), `-`(`/`...
proc (n) options operator, arrow; `+`(`-`(`/`(25, 32)), `*`(`/`(1, 4), `*`(`^`(6, `/`(1, 2)), `*`(`^`(n, `/`(1, 2))))), `/`(`*`(`/`(93, 1024), `*`(`^`(6, `/`(1, 2)))), `*`(`^`(n, `/`(1, 2)))), `-`(`/`...
proc (n) options operator, arrow; `+`(`-`(`/`(25, 32)), `*`(`/`(1, 4), `*`(`^`(6, `/`(1, 2)), `*`(`^`(n, `/`(1, 2))))), `/`(`*`(`/`(93, 1024), `*`(`^`(6, `/`(1, 2)))), `*`(`^`(n, `/`(1, 2)))), `-`(`/`...
(6.1.1.5)
 

Initial approximants Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( 

> m := 1;
 

1 (6.1.2.1)
 

> `u'` := unapply( add( tau[j]/n^(j/2), j=-4..m-1 ), n );
 

proc (n) options operator, arrow; `+`(`-`(`/`(25, 32)), `*`(`/`(1, 4), `*`(`^`(6, `/`(1, 2)), `*`(`^`(n, `/`(1, 2)))))) end proc (6.1.2.2)
 

> `delta'`[n] = asympt( `u'`(n) - u(n) , n, 2 );
 

`delta'`[n] = `+`(`-`(`*`(`/`(93, 1024), `*`(`^`(6, `/`(1, 2)), `*`(`^`(`/`(1, `*`(n)), `/`(1, 2)))))), `/`(`*`(`/`(5575, 4096)), `*`(n)), `-`(`*`(`/`(23263, 524288), `*`(`^`(6, `/`(1, 2)), `*`(`^`(`/... (6.1.2.3)
 

New approximants Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( 

> `phi'` := n -> 1 + m/(2*n);
`psi'` := n -> ' `a'`(n)*a(n+1) / (a(n+1)+`b'`(n)*b(n+1) + `b'`(n)*`u'`(n+1))^2 ';
 

 

proc (n) options operator, arrow; `+`(1, `/`(`*`(`/`(1, 2), `*`(m)), `*`(n))) end proc
proc (n) options operator, arrow; '`/`(`*`(`a'`(n), `*`(a(`+`(n, 1)))), `*`(`^`(`+`(a(`+`(n, 1)), `*`(`b'`(n), `*`(b(`+`(n, 1)))), `*`(`b'`(n), `*`(`u'`(`+`(n, 1))))), 2)))' end proc (6.1.3.1)
 

> `u''` := unapply( (`phi'`(n)*(`a'`(n)/( `b'`(n)+a(n+1)/(b(n+1)+`u'`(n+1)) )) - `psi'`(n)*`u'`(n))/(`phi'`(n) - `psi'`(n)) , n):
`u''`[n] = asympt( `u''`(n), n, 5 );
 

`u''`[n] = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(6, `/`(1, 2)))), `*`(`^`(`/`(1, `*`(n)), `/`(1, 2)))), `-`(`/`(25, 32)), `*`(`/`(93, 1024), `*`(`^`(6, `/`(1, 2)), `*`(`^`(`/`(1, `*`(n)), `/`(1, 2))))), `-`(...
`u''`[n] = `+`(`/`(`*`(`/`(1, 4), `*`(`^`(6, `/`(1, 2)))), `*`(`^`(`/`(1, `*`(n)), `/`(1, 2)))), `-`(`/`(25, 32)), `*`(`/`(93, 1024), `*`(`^`(6, `/`(1, 2)), `*`(`^`(`/`(1, `*`(n)), `/`(1, 2))))), `-`(...
(6.1.3.2)
 

> `delta''`[n] = asympt( `u''`(n) - u(n) , n, 5 );
 

`delta''`[n] = `+`(`-`(`*`(`/`(35951, 524288), `*`(`^`(6, `/`(1, 2)), `*`(`^`(`/`(1, `*`(n)), `/`(3, 2)))))), O(`/`(1, `*`(`^`(n, 2))))) (6.1.3.3)
 

> m := m+2;
 

3 (6.1.2)
 

> #`phi'`(n);
#`psi'` := n -> ' `a'`(n)*a(n+1) / (a(n+1)+`b'`(n)*b(n+1) + `b'`(n)*`u''`(n+1))^2 ';
 

>
 

> `u'''` := unapply( (`phi'`(n)*( `a'`(n)/( `b'`(n)+a(n+1)/(b(n+1)+`u''`(n+1)) ) ) - `psi'`(n)*`u''`(n))/(`phi'`(n) - `psi'`(n)), n ):
 

> asympt( `u'''`(n) - u(n) , n, 7 );
 

`+`(`*`(`/`(5104771, 134217728), `*`(`^`(6, `/`(1, 2)), `*`(`^`(`/`(1, `*`(n)), `/`(5, 2))))), `-`(`/`(`*`(`/`(32952821, 16777216)), `*`(`^`(n, 3)))), O(`*`(`^`(`/`(1, `*`(n)), `/`(7, 2))))) (6.1.3)
 

> m := m+2;
 

5 (6.1.4)
 

> `u''''` := unapply( (`phi'`(n)*( `a'`(n)/( `b'`(n)+a(n+1)/(b(n+1)+`u'''`(n+1)) ) ) - `psi'`(n)*`u'''`(n))/(`phi'`(n) - `psi'`(n)), n ):
 

> asympt( `u''''`(n) - u(n) , n, 8 );
 

`+`(`*`(`/`(3743217385, 34359738368), `*`(`^`(6, `/`(1, 2)), `*`(`^`(`/`(1, `*`(n)), `/`(7, 2))))), O(`/`(1, `*`(`^`(n, 4))))) (6.1.5)
 

>