I have the following series of reals: $$ \sum_{j=0}^\infty\sum_{l=0}^\infty \frac{\Gamma\left(m+j+l+\frac12\right) x^j y^{j+l}}{j!l!(k+j+l)!\Gamma(m+j+\frac12) }, $$ Here $k$ is a positive integer and $m$ is a nonnegative integer. (I am most interested in the case where $k=1$ and $m=0,1$, but would ideally love to know for the general case.)
I can show that my series converges for $x$ and $y$ that I need for my case, but $x$ and $y$ can both be greater than 1.
I think that my series reduces to
$$
\frac1{(k-1)!} \sum_{j=0}^\infty\sum_{l=0}^\infty \frac{ \left(m+\frac12\right)_{j+l} x^j y^{l+j}}{j!l!(k)_{j+l}\left(m+\frac12\right)_j},
$$
where $(a)_n$ is the Pochhammer symbol, as per https://mathworld.wolfram.com/PochhammerSymbol.html
Assuming that my reduction above is correct, this appears to be the Kampé de Fériet function in two variables from
https://en.wikipedia.org/wiki/Kamp%C3%A9_de_F%C3%A9riet_function
I had never heard of this function before, and I was wondering how to compute it while avoiding calculating the series and sums.
A similar question was asked at How to calculate the Kampé de Fériet function? where the solution seems to be in terms of $x=y$ (and in particular equal to 1). I am looking for an expression on how to calculate or reduce it for $x$ and $y$ both being distinct.
Thanks in advance for any suggestions or approaches!
