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Euler gave the following well-known integral representations for the Gauss hypergeometric function ${_2F_1}$ and the generalized hypergeometric function ${_3F_2}$: for $0<\Re{\left(\beta\right)}<\Re{\left(\gamma\right)}$,

$$\small{{_2F_1}{\left(\alpha,\beta;\gamma;z\right)}=\frac{1}{\operatorname{B}{\left(\beta,\gamma-\beta\right)}}\int_{0}^{1}\frac{t^{\beta-1}\left(1-t\right)^{\gamma-\beta-1}}{\left(1-zt\right)^{\alpha}}\,\mathrm{d}t};\tag{1}$$

and for $0<\Re{\left(\mu\right)}<\Re{\left(\nu\right)}$,

$$\small{{_3F_2}{\left(\alpha,\beta,\mu;\gamma,\nu;z\right)}=\frac{1}{\operatorname{B}{\left(\mu,\nu-\mu\right)}}\int_{0}^{1}t^{\mu-1}\left(1-t\right)^{\nu-\mu-1}{_2F_1}{\left(\alpha,\beta;\gamma;zt\right)}\,\mathrm{d}t}.\tag{2}$$

I'm curious to learn if there is a way evaluate the following integral (possibly in terms of higher order generalized hypergeometric functions or the two-variable Appell functions?):

$$\small{\mathcal{I}{\left(\alpha,\beta,\gamma,z;\mu,\nu,\rho,w\right)}=\int_{0}^{1}\frac{t^{\mu-1}\left(1-t\right)^{\nu-\mu-1}}{\left(1-wt\right)^{\rho}}{_2F_1}{\left(\alpha,\beta;\gamma;zt\right)}\,\mathrm{d}t}.\tag{3}$$

Now, this integral $\mathcal{I}$ is a straightforward generalization of $(3)$, and it seems only natural to me that there is a paper on this integral out there somewhere. But if it exists it has eluded me, despite most furious Googling on my part.

If any of our resident master integrators have any insight to offer, I'd be very grateful. I'd also welcome any niche references that might be relevant here if someone happens to have any.

Cheers!

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In the special case $\mu=\gamma$ Prudnikov-Brychkov-Marychev (Vol. III, formula 2.21.1.20) gives an evaluation in terms of Appell's $F_3$ (of four Appel's functions, this is the one with the maximal number of parameters): $$\mathcal I\left(\alpha,\beta,\gamma,z;\gamma,\nu,\rho,w\right)= \frac{B\left(\gamma,\nu-\gamma\right)}{(1-w)^{\rho}}{}F_3\left(\rho,\alpha,\nu-\gamma,\beta,\nu;\frac{w}{w-1};z\right)$$ Since all arguments are ''free'' (there is no relation between them), your generalization is yet one more step beyond Appell. A relation to generalized hypergeometric functions for generic $w,z$ would be extremely surprising.

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  • $\begingroup$ Thank you very much for your response. The only reason I hesitate to accept it is that you leave open the question about whether the general case is unsolvable or just merely incredibly difficult. That said, the special case you give does in fact suffice for my underlying interest in this question. If no other responses appear in the next couple of days, then I'll accept this answer. $\endgroup$ – David H Dec 27 '15 at 6:11
  • $\begingroup$ Specifically, I'm very interested in the special case $F_3{\left(\frac12,\frac12,\frac12,\frac12,2;x;y\right)}$ and whether it can be reduced to elliptic integrals or the Appell $F_1$ function. If you can help me with this special case, I'll not only accept your response, I'd be inclined to award it the maximum bounty. $\endgroup$ – David H Dec 27 '15 at 6:18
  • $\begingroup$ Dear @DavidH sorry that the answer sounds negative. It might happen that there exists some further generalization of Appell's functions which incorporates your integral (Kampé de Fériet or maybe some multivariate hypergeometrics?) but I am afraid that in any case it will be not more than a notation. As for the specific $F_3$ from your comment, does it come from something like $\int_0^1 \frac{ \arcsin \sqrt{a t} \,dt}{ \sqrt{ (1-t) (1-bt) }}$ ? $\endgroup$ – Start wearing purple Dec 27 '15 at 10:31
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    $\begingroup$ Indeed, it comes from an integral I asked about here many months ago. I've found that this integral is the lynchpin in obtaining a closed form expression for a wide class of integrals I've encountered in certain mathematical physics problems, hence my interest. $\endgroup$ – David H Dec 27 '15 at 11:00
  • $\begingroup$ How $B(\gamma,\nu-\gamma)\mathrm{F}^{1:2;1}_{1:1;0}\Bigg(\begin{matrix}\gamma&:&\alpha,\beta&;&\rho&\\\nu&:&\gamma&;&-&\end{matrix}\Bigg|z,w\Bigg)$ can reduces to $\dfrac{B\left(\gamma,\nu-\gamma\right)}{(1-w)^{\rho}}F_3\left(\rho,\alpha,\nu-\gamma,\beta,\nu;\dfrac{w}{w-1};z\right)$ ? $\endgroup$ – Harry Peter Aug 30 '16 at 12:05
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$\int_0^1\dfrac{t^{\mu-1}(1-t)^{\nu-\mu-1}}{(1-wt)^\rho}{_2F_1}(\alpha,\beta;\gamma;zt)~dt$

$=\int_0^1\sum\limits_{n=0}^\infty\dfrac{(\alpha)_n(\beta)_nz^nt^{n+\mu-1}(1-t)^{\nu-\mu-1}}{(\gamma)_nn!(1-wt)^\rho}dt$

$=\sum\limits_{n=0}^\infty\dfrac{(\alpha)_n(\beta)_nz^nB(n+\mu,\nu-\mu){_2F_1}(\rho,n+\mu;n+\nu;w)}{(\gamma)_nn!}$

$=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^\infty\dfrac{\Gamma(n+\mu)\Gamma(\nu-\mu)(\alpha)_n(\beta)_n(\rho)_k(n+\mu)_kz^nw^k}{\Gamma(n+\nu)(\gamma)_n(n+\nu)_kn!k!}$

$=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^\infty\dfrac{\Gamma(n+k+\mu)\Gamma(\nu-\mu)(\alpha)_n(\beta)_n(\rho)_kz^nw^k}{\Gamma(n+k+\nu)(\gamma)_nn!k!}$

$=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^\infty\dfrac{\Gamma(\mu)\Gamma(\nu-\mu)(\mu)_{n+k}(\alpha)_n(\beta)_n(\rho)_kz^nw^k}{\Gamma(\nu)(\nu)_{n+k}(\gamma)_nn!k!}$

$=B(\mu,\nu-\mu)\mathrm{F}^{1:2;1}_{1:1;0}\Bigg(\begin{matrix}\mu&:&\alpha,\beta&;&\rho&\\\nu&:&\gamma&;&-&\end{matrix}\Bigg|z,w\Bigg)$

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  • $\begingroup$ Ok, I've seen hypergeometric functions using notation similar to the one you give in the last line above, but I'm having trouble finding a definition of what the indices represent. Can you explain, or redirect me to a complete definition? $\endgroup$ – David H Aug 29 '16 at 4:50

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