Let $C_k := \frac{1}{k + 1} \binom{2k}{k}$ be the $k$-th Catalan number and let $K$ be a positive integer.
I am looking for an identity or simplification of
\begin{equation} \sum_{k = 0}^K C_k \binom{K + k}{K - k} z^k \end{equation}
that does not involve a summation. Mathematica returns the ordinary hypergeometric function $_2F_1(-K,K + 1,2,-z)$, but that still involves a summation. Can anyone point me to a reference that could show me the right direction?