adm_quad.rr

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Manual

This program seeks admissible quadruples (a,b,c,x) (see Section 1 in [1]) for a given triple of integers (k,l,m) with

0 <= k+l-m <= l-k <= m.

By finding an admissible quadruple (a,b,c,x) for (k,l,m), we obtain the degenerate relation

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

(see (1.7) in Section 1 in [1]). From this relation, we can obtain special values of 2F1(a,b;c;x).

If we input

>adm_quad(k,l,m);

, then polynomial systems which admit admissible quadruples for (k,l,m) are outputted.

For example, if we input

>adm_quad(1,2,3);

, then output is

>[[3*b-2*c,3*a-c-1,x-9],[3*b-2*c+1,3*a-c,x-9]]

This means that the admissible quadruples for (k,l,m)=(1,2,3) are (a,b,c,x)=(a,2a-1/3,3a,9), (a,2a-2/3,3a-1,9) (see 4.3.2 in [1]).

By commanding

>gauss_3tr_R(a,2*a-1/3,3*a,1,2,3,9);

of the Risa/Asir program "3tr.rr", we can get R(x) satisfying

2F1(a+1,2a+5/3;3a+3;9)=R(x)*2F1(a,2a-1/3;3a;9).

References

[1] A. Ebisu, Special values of the hypergeometric series, preprint.

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