Generalized Cesàro summability of $(-1)^nn^p$

Question

A sequence $\{a_n\}_{n\geq 0}$ is said to be $(C,\alpha)$-summable if $\lim_{n\to\infty} S^\alpha_n$ exists, where $$ S^n_\alpha = \sum_{k=0}^n \frac{ {n \choose k} }{ {n+\alpha \choose k} } a_k. $$ How could one show $a_k = (-1)^k(2k+1)^2$ is $(C,3)$-summable? In general, is it true that the sequence $a_k = (-1)^k k^p$ is $(C,p+1)$-summable for positive integers $p$?