Reduce the product of two hypergeometric functions to a single hypergeometric function [closed]

I have this product of two hypergeometric functions

Hypergeometric2F1[2 + 3 d5 + 2 k5, 2 + 3 d5 + 2 k5, 4 + 6 d5 + 4 k5, 1 - 1/t^2]
HypergeometricPFQ[{d5/2, d5, -(d5/2) - k5}, {1 + d5/2, 1 + (3 d5)/2 + k5}, t^2]

to which I want to apply the conversion rule posted here to a single hypergeometric function, using (as suggested by a comment of Johannes Trost) the values $z=1,c=t^2,d=1-\frac{1}{t^2}$. (Feel free to suggest other values.) I've been trying to do so, and am not sure I have been fully successful--in particular, I've not been certain as to the import/consequence of the Condition operator ( /; ) at the end of the summation. Also, my attempts to perform the indicated infinite summation have not produced any significant results--which may be unavoidable, due to the difficulty of the underlying problem.

In any case, I hope to perform an integration over $t \in [0,1]$ incorporating this hypergeometric product or its equivalent as discussed in this post.

(I changed the $d,k$ symbols in that posting to $d5,k5$ to avoid conflict of symbols.)

Here are the variable/parameter settings I have been using.

r = 2; s = 1; p = 3; q = 2; z = 1; c = t^2; d = 1 - 1/t^2;
{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]} = {d5/2, d5, -(d5/2) - k5};
{Subscript[b, 1], Subscript[b, 2]} = {1 + d5/2, 1 + (3 d5)/2 + k5};
Subscript[α, 1] = 2 + 3 d5 + 4 k5;
Subscript[α, 2] = 2 + 3 d5 + 4 k5;
Subscript[β, 1] = 4 + 6 d5 + 4 k5;

closed as unclear what you're asking by Sektor, halirutan♦, MarcoB, Coolwater, Henrik SchumacherMay 28 '18 at 7:38

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Please, post a self-contained snippet of code for others to play with. If I have to piece together something that supposedly works from 4 different places I would be discouraged to say the least. – Sektor May 25 '18 at 20:40
• Variable/parameter settings for what? Pretend that you know nothing about hypergeometric functions, but you are a Mathematica power user. Now, read your Q. Would you be able to piece the information sufficient to solve the problem? Doubt it! Describe succinctly what you have as a given, your assumptions and what you are trying to prove/compute. – Sektor May 25 '18 at 22:28
• As a suggestion -- Avoid Subscript and Superscript when using Mathematica. Please, refer to the help centre on how to properly format your Qs. – Sektor May 25 '18 at 22:30