Find region of convergence of double power series

Question

How can i calculate the region of convergence of this double power series ? $$ S(x,y)=\displaystyle{ \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{(n-\frac12)!\>(k-\frac12)!\>(\frac{n}{2}+k-\frac34)!}{(\frac{n}{2}-\frac34)!\>(k+\frac12)!}}\frac{x^{n}\>y^{n}}{n!\>k!} $$ Looking aroundi found the citation of the generalized Cauchy-Hadamard theorem but no way to apply it. Thanks in advance

Answer 1

The series in question is a hypergeometric series in 2 variables and the region convergence can be found using Horn's theorem. Applying Horn's theorem would give the region of convergence as $| x| <1\land | y| <1\land | y| <1-| x| ^2$.

One can also use automatized Mathematica package which can be found here: https://arxiv.org/abs/2201.01189. Horn's theorem is also mentioned in the paper.