# On a hypergeometric-type integral

I'm having a bit of trouble with the integral $$\int_0^1 e^{-\frac{z^2}{2}u}\frac{u^{m-\frac{1}{2}} (1-u)^{n/2} }{ \left(1+ \left(s^2-1\right)u\right)^{m+\frac{n}{2}+1}} du.$$ (Here $m$ and $n$ are nonnegative integers, $s>0$ and I can assume that $z\in\mathbb{R}$.) For $s=0$ it reduces to the standard integral representation of the confluent hypergeometric function ${}_1F_1$, which is great. I would like to extend it to the case $s>0$, which then looks very much like many integrals in Gradshteyn, Erdélyi, Prudnikov and the DLMF, but I can't quite match it to anything.

The reason I ask here is that this is reducible to the form $$\int_0^1 u^{a-1} (1-u)^{b-a-1}{}_0F_0(-;-;-\zeta u)\times{}_1F_0(a';-;s'u),$$ which is the Euler type of integral that's apparently so common, but I can't find a general enough form for this.

I know this is a bit of a long shot but I'm hoping someone with more intuition for hypergeometric functions will be able to point me in the right direction.

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"$\sigma=0$" should presumably be "$s=0$". –  Noam D. Elkies Mar 12 at 22:55
Yes, it's fixed now. –  Emilio Pisanty Mar 13 at 13:27