More congruences from Apery-like formulae

TL;DR

This paper derives new congruences modulo high powers of primes for specific binomial sum series involving powers of the index, extending known results in number theory.

Contribution

It introduces novel congruences modulo p^{6-d} for sums involving binomial coefficients and powers, generalizing Apery-like formulas.

Findings

Established congruences modulo p^{6-d} for sums with binomial coefficients.

Extended the class of known Apery-like congruences.

Provided formulas for sums with different powers d=1,2,3.

Abstract

We present some congruences modulo $p^{6-d}$ for sums of the type $\sum_{k=0}^{(p-3)/2}x^k{2k\choose k}/(2k+1)^d$, for $d=1,2,3$ where $p>5$ is a prime.