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

TL;DR

This paper explores congruences related to Apéry-type series for zeta values, extending known results to higher zeta values and confirming conjectures through p-analogues and binomial sum analyses.

Contribution

It introduces new congruences for truncated Apéry-type series for b6(4) and b6(5), and proves a p-analogue of Zeilberger's series for b6(2), confirming Sun's conjecture.

Findings

Established congruences for finite sums related to b6(4) and b6(5).

Proved a p-analogue of Zeilberger's series for b6(2).

Confirmed a conjecture of Z. W. Sun.

Abstract

Recently, R. Tauraso established finite $p$-analogues of famous Ap\'ery series for $\zeta(2)$ and $\zeta(3).$ 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 $\zeta(5).$ We also prove a $p$-analogue of Zeilberger's series for $\zeta(2)$ confirming a conjecture of Z. W. Sun.