Congruences arising from Ap\'ery-type series for zeta values

Recently, R. Tauraso established finite $p$-analogues of famous Ap'ery series for $\zeta (2)$ and $\zeta (3)\mathrm{.}$ In this paper, we present several congruences for finite central binomial sums arising from the truncation of Ap'ery-type series for $\zeta (4)$ and $\zeta (5)\mathrm{.}$ We also prove a $p$-analogue of Zeilberger’s series for $\zeta (2)$ confirming a conjecture of Z. W. Sun.