Evaluating a certain integral which generalizes the ${_3F_2}$ hypergeometric function

Question

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!

Answer 1

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.

Answer 2

$\int_0^1\dfrac{t^{\mu-1}(1-t)^{\nu-\mu-1}}{(1-wt)^\rho}{_2F_1}(\alpha,\beta;\gamma;zt)~dt$

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

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

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

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

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

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