Congruences concerning Jacobi polynomials and Apéry-like formulae

Abstract

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

to appear in Int. J. Number Theory

References in corpus (1)

• Congruences concerning Legendre polynomials

Cited by in corpus (1)

• Supercongruences involving products of two binomial coefficients modulo $p^4$