Show that $$\sum_{k=0}^{\infty}\sum_{n=0}^{\infty}[k|\beta +1]_n\frac{|t|^k}{k!}\frac{|x|^n}{n!}$$ is convergent $\forall t \in \mathbb{R}$, and $x, \beta \in \mathbb{R}$ such as $|\beta x|<1$, where $[k|\beta +1]_n=k(k+\beta)(k+2\beta)...(k+(n-1)\beta)$. By definition, $[k|\beta +1]_0=1$.
I don´t have so much theory about double series, my idea is bounding or see the radius of convergence. I really thank if somebody give me other idea .
It suffices to see the convergence for $\beta\geqslant 0$ (it is then absolutely convergent for any $\beta$); as each term is nonnegative in this case, we can consider the iterated summation. The inner sum (over $n$) converges to $(|t|^k/k!)(1-\beta|x|)^{-k/\beta}$ (as a binomial series), therefore the entire sum converges to $$\exp\big(|t|(1-\beta|x|)^{-1/\beta}\big).$$ (Thus actually we've computed the sum for any $\beta$.)