# Integrals of product of hypergeometric functions

1. Apr 8, 2010

### Pere Callahan

Hello,

I am wondering about integrals of the form

$$\int_0^1 {}_p F_q(\{a_1,\ldots,a_p\},\{b_1,\ldots,b_q\},y){}_{p'} F_{q'}(\{a'_1,\ldots,a'_{p'}\},\{b'_1,\ldots,b'_{q'}\},y)dy$$

integrals of product of hypergeometric functions.

I know that if the limits of integration were +/- infty the convolution property of Meijer-G functions would give an answer to that question. I also know that for the special case of Bessel functions there are formulas known by Mathematica.

If at least one of the hypergeometric functions, however, is more complicated, Mathematica reaches its limits.

My question is if anyone knows where I could fine more informaiton on this (yes, I checked the handbooks of special functions, and tables of integrals).

Thanks a lot

Pere

2. Apr 10, 2010

### Pere Callahan

So maybe I should phrase the question more explicitly. Using recursion formulas and identities for contiguous functions, I am able to reduce the general integrals to
$$\int_0^1 {\operatorname{J}_1(2y) {}_2\operatorname{F}_3(\{1,1\},\{2,2,2\},y)dy}.$$

As I said, Mathematica and similar programs seem to be unable to express this integrals in terms of known special functions but this does not convince me that it is impossible.

Any input is gratefully appreciated,

pere