I am attempting to find an analytic solution for the following infinite sum, where $x$ is a nonnegative real number:
\begin{equation} \sum_{k=1}^{\infty} \frac{(-1)^k y^{2k} k}{(2k)!} \, {}_1F_2\left(k+1;2,k+\frac{1}{2};-\frac{x^2}{4}\right) \end{equation}
The series involves a generalized hypergeometric function $ {}_1F_2 $ with these specific parameters. While I have come across integral representations that involve $ {}_0F_2 $ hypergeometric functions, they haven't led to any useful simplification of the sum.
My main objective is to find an analytic solution to this infinite sum. Does anyone know of a specific integral representation, identity, or transformation for this $ {}_1F_2 $ function that would facilitate solving the sum analytically?
Any pointers, insights, or references would be highly appreciated.