This is my first time asking a question in Math SE. During my research I am interested in evaluating this specific integral:
\begin{equation} L(\tau) = \int_{0}^{\infty}r^{n_c-1}e^{-r}\left[F_{2}\left(1,0,0;1,n_{c};\frac{p}{1+\tau},\frac{qr}{1+\tau}\right)\right]^{1+\tau}dr. \end{equation}
Here, $n_{c}\in\mathbb{N},~p,q\in\mathbb{R^+},~\tau\in[0,1]$, and $F_2$ is an Appell hypergeometric function of two variables defined as \begin{equation} F_{2}(\alpha,\beta_1,\beta_2;\gamma_1,\gamma_2;x,y) = \sum_{m,n=0}^{\infty}\frac{(\alpha)_{m+n}(\beta_1)_m(\beta_2)_n}{(\gamma_1)_m(\gamma_2)_nm!n!}x^{m}y^{n} \end{equation} where $(a)_k = a(a+1)\cdots(a+k-1)$ is a Pochhammer symbol. I only managed to derived a closed-form expression for $\tau=0$ but had no luck with other values of $\tau$.
Here is my attempt at special case $\tau=0$. Observe that \begin{eqnarray} F_{2}\left(1,0,0;1,n_{c};\frac{p}{1+\tau},\frac{qr}{1+\tau}\right) &=& \sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{p}{1+\tau}\right)^{m}\sum_{n=0}^{\infty}\frac{(m+n)!}{m!(n_c)_{n}n!}\left(\frac{qr}{1+\tau}\right)^{n}\\ &=& \sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{p}{1+\tau}\right)^{m}\sum_{n=0}^{\infty}\frac{(m+1)_{n}}{(n_c)_{n}n!}\left(\frac{qr}{1+\tau}\right)^{n}\\ &=& \sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{p}{1+\tau}\right)^{m}{}_1F_1\left(m+1;n_c;\frac{qr}{1+\tau}\right) \end{eqnarray} where $_1F_1(;;)$ is the Kummer hypergeometric function. Now, \begin{eqnarray} L(0) &=& \sum_{m=0}^{\infty}\frac{p^{m}}{m!}\int_{0}^{\infty}r^{n_c-1}e^{-r}{}_1F_1(m+1;n_c;qr)dr \end{eqnarray} Using the identity (from https://arxiv.org/pdf/math-ph/0306043.pdf, equation (2.4)) \begin{equation} \int_{0}^{\infty}r^{d-1}e^{-ht}{}_1F_1(a;b;kt)dt = h^{-d}\Gamma(d)_2F_1\left(d,a;b;\frac{k}{h}\right) \end{equation} where ${}_2F_1(;;)$ is the Gauss hypergeometric function, we have \begin{eqnarray} L(0) &=& \Gamma(n_c)\sum_{m=0}^{\infty}\frac{p^{m}}{m!}{}_2F_1\left(n_c,m+1;n_c;q\right)\\ &=& \Gamma(n_c)\sum_{m,n=0}^{\infty}\frac{(n_c)_{n}(m+1)_{n}}{(n_c)_{n}m!n!}p^{m}q^{n}\\ &=& \Gamma(n_c)\sum_{m,n=0}^{\infty}\frac{(m+n)!}{m!m!n!}p^{m}q^{n}\\ &=& \Gamma(n_c)F_{2}\left(1,0,0;1,0;p,q\right) \end{eqnarray} The last one is the desired expression. But I am having trouble for deriving expression for $0<\tau\leq 1$. Thank you very much in advance.