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I am wondering whether there exists a closed form for the definite integral $$F(x)=\int_0^1t^{-a}(1-t)^{N}(1-xt)^{-a}{}_2F_1(-a,k-a-1/2,k-a;4xt(1-xt))dt,$$ where $a\in(0,1)$ and $N,k$ are positive integers.

It seems that I cannot apply Kummer's quadratic transformation formula directly. So I tried to follow the method of proving Kummer's formula by expanding $_2F_1$, and I require some formula for $_2F_1(A,-A,C;x)$. However I cannot find anything relevant in many literatures.

Do you have some other ideas?

Thanks in advance.

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2 Answers

May be use the series for $_{2}F_{1}$ and then integrate term by term? The result looks like a sort of Appell-type function of two arguments: $x, 1-x$...

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The Maple 18 code $$ A := eval(convert(hypergeom([-a, k-a-1/2], [k-a], y), FPS, y), y = 4*x*t*(-t*x+1))\,assuming k::posint:$$ $$int(A*t^{-a}*(1-t)^N*(-t*x+1){-a}, t = 0 .. 1)\,assuming a>0,a<1,N::posint,x>0,x<1 $$ outputs the integral under consideration as the series $$\Gamma(N+1)\times$$ $$\sum\limits_{k1=0}^\infty \frac{\mathop{\rm pochhammer}(-a,k1)x^{k1}2^{2k1}{\mbox{$_2$F$_1$}(- k1+a,1-a+ k1;-a+ k1+2+N;x)}}{\mathop{\rm pochhammer}(k-a,k1)\Gamma(-a+k1+2+N)\mathop{\rm pochhammer}(k-a-1/2,k1)^{-1}k1!},$$ where $\mathop{\rm pochhammer}$ is described here. See here for the Maple output.

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