Generating relation for Bateman F

Theorem

The following formula holds: $$\displaystyle\sum_{k=0}^{\infty} F_n(z)t^n = \dfrac{1}{1-t} {}_2F_1 \left( \dfrac{1}{2}, \dfrac{1+z}{2}; 1; \dfrac{-4t}{(1-t)^2} \right),$$ where $F_n$ denotes the Bateman F and ${}_2F_1$ denotes the hypergeometric pFq.