Congruences concerning Jacobi polynomials and Ap\'ery-like formulae

TL;DR

This paper establishes new congruences modulo powers of primes for sums involving binomial coefficients and powers, extending known results related to Jacobi polynomials and Apéry-like formulas.

Contribution

It proves novel congruences for specific binomial sum forms modulo higher powers of primes, generalizing previous results in number theory.

Findings

Congruences modulo p^{3-d} for sums with binomial coefficients and powers.

Enhanced congruences modulo p^{5-d} for special parameter cases.

Extension of Apéry-like formulae in the context of prime moduli.

Abstract

Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case $t=(-1)^{d}/16$ of the former sum, where the congruences hold modulo $p^{5-d}$.