Is it possible to calculate this integral ? $$I:=\int_0^{\gamma^2} \int_0^{\gamma^2} \left((1-x)(1-y)\right)^{s-2} {}_3F_2\left(s,s,s;1,1;xyt\right) dx dy$$ where $\gamma,\; t,\; s\geq 0$ and ${}_3F_2$ is the hypergeometric series.
Although the integral looks neat and fairly simple in form. I tried evaluating it using integration by parts but without success. I also couldn't find the solution in the book, "Table of Integrals, Series, and Products" by Gradshteyn and Ryzhik and Prudnikov, Brychkov, - Integrals and Series 1-3.
Can anyone help me in solving this?
Thank you.
I do not know about a solution for arbitary $\gamma$, but for the special case of $\gamma=\pm 1$ the integral in question reduces to $$ I=\int_0^1\int_0^1((1-x)(1-y))^{s-2} {}_3F_2\left({s,s,s\atop 1,1};xyt\right)\,\mathrm dx\mathrm dy. $$ Through the use of DLMF 16.5.2 we may identify the integral w.r.t. $x$ using $a_0=1$ and $b_0=s$ to obtain $$ I=\frac{1}{s-1}\int_0^1(1-y)^{s-2} {_4F_3}\left({s,s,s,s\atop 1,1,1};yt\right)\,\mathrm dy. $$ Note that the use of the DLMF integral representation requires $\Re s>1$. Application of DLMF 16.5.2, again for $a_0=1$ and $b_0=s$, then gives the final solution $$ I=\frac{1}{(s-1)^2}{_5F_4}\left({s,s,s,s,s\atop 1,1,1,1};t\right). $$