3tr.rr

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Manual

This program rapidly calculates the three term relation of the hypergeometric series 2F1. Namely, for integers k, l, m, this program computes Q(x), R(x) satisfying

2F1(a+k,b+l;c+m;x)=Q(x)2F1(a+1,b+1;c+1;x)+R(x)2F1(a,b;c;x).

This also calculates q(x), r(x) satisfying

(a,k)*(b,l)/(c,m)*2F1(a+k,b+l;c+m;x)=a*b/c*q(x)*2F1(a+1,b+1;c+1;x)+r(x)2F1(a,b;c;x),

where (a,k) is the pochhammer symbol.

If you want q(x) with free parameters a,b,c and x for a triple (k,l,m), then input

>gauss_3tr_q(a,b,c,k,l,m,x);

in Risa/Asir.

Similarly, if you want r(x) with free parameters a,b,c and x for a triple (k,l,m), then input

>gauss_3tr_r(a,b,c,k,l,m,x);

If a,b,c,x have some relations, for example, (a,b,c,x)=(a,1-a,c,1/2), then input

>gauss_3tr_q_red(a,1-a,c,k,l,m,1/2);

for obtaining q(x) with (a,b,c,x)=(a,1-a,c,1/2), and

>gauss_3tr_r_red(a,1-a,c,k,l,m,1/2);

for obtaining r(x) with (a,b,c,x)=(a,1-a,c,1/2).

If you want Q(x) and R(x), then input

>gauss_3tr_Q(a,b,c,k,l,m,x);

and

>gauss_3tr_R(a,b,c,k,l,m,x);

, respectively, even if a,b,c,x have some relations.

Remark:

If a,b,c have the relations such that

a-b is a integer

or

c is a integer,

for example (a,b,c)=(1/2,1/2,1), then outputs may be incorrent.

For avoiding such situations, please substitute (a,b,c,x) for the result of general case.

For example, input subst(gauss_3tr_q(a,b,c,k,l,m,x),a,1/2,b,1/2,c,1);

Reference: A. Ebisu, Three term relations for the hypergeometric series, Funkcialaj Ekvacioj, 55(2012), 255-283.