I have a hunch that the following sum can be expressed as a single $G$-function: $$ S(t,y) = \sum_{k=0}^\infty \frac{t^{2k}}{(2k)!} G_{p+2,q+2}^{m+2,n}\left(y\left|\begin{smallmatrix}\vec{a}-k,-k,-k-\frac{1}{2}\\ \newline 0,-\frac{1}{2},\vec{b}-k-\frac{1}{2}\end{smallmatrix}\right.\right) $$
where $\vec{a}$ and $\vec{b}$ are $p$ and $q$ dimensional vectors of parameters respectively.
Using the identity
$$ (-y)^k \frac{d^k}{dy^k}G_{p,q}^{m,n}\left(y\left|\begin{smallmatrix}\vec{a}\\ \newline \vec{b}\end{smallmatrix}\right.\right) = G_{p+1,q+1}^{m+1,n}\left(y\left|\begin{smallmatrix}\vec{a},0\\ \newline k,\vec{b}\end{smallmatrix}\right.\right) $$ I noticed that I can write the $G$-function as
$$ G_{p+2,q+2}^{m+2,n}\left(y\left|\begin{smallmatrix}\vec{a}-k,-k,-k-\frac{1}{2}\\ \newline 0,-\frac{1}{2},\vec{b}-k-\frac{1}{2}\end{smallmatrix}\right.\right) = (-1)^k \frac{d^k}{dy^k}G_{p+1,q+1}^{m+1,n}\left(y\left|\begin{smallmatrix}\vec{a},-\frac{1}{2}\\ \newline k-\frac{1}{2},\vec{b}\end{smallmatrix}\right.\right) $$
This is close to making $S(t,y)$ look like a Taylor series, but there is still a $k$ left over, so I can't do the sum. Trying the same trick again results in
$$ G_{p+2,q+2}^{m+2,n}\left(y\left|\begin{smallmatrix}\vec{a}-k,-k,-k-\frac{1}{2}\\ \newline 0,-\frac{1}{2},\vec{b}-k-\frac{1}{2}\end{smallmatrix}\right.\right) = \frac{d^k}{dy^k}\left[y^{k-\frac{1}{2}}\frac{d^k}{dy^k}G_{p,q}^{m,n}\left(y\left|\begin{smallmatrix}\vec{a}+\frac{1}{2}\\ \newline \vec{b}+\frac{1}{2}\end{smallmatrix}\right.\right)\right] $$
which just seems even further from something useful.
Are there any tricks I might be missing that would help me to simplify this?