Congruences involving $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$

TL;DR

This paper establishes new congruences involving the sequence g_n(x), related to Apéry and Franel numbers, demonstrating properties similar to Wolstenholme's classical congruences for primes greater than 5.

Contribution

The paper proves fundamental congruences involving g_n(x), extending known results and revealing new properties of these sequences in number theory.

Findings

For primes p > 5, sum of g_k(-1)/k over k=1 to p-1 is divisible by p^2.

For primes p > 5, sum of g_k(-1)/k^2 over k=1 to p-1 is divisible by p.

The results are analogous to Wolstenholme's classical congruences.

Abstract

Define $g_n(x)=\sum_{k=0}^n\binom nk^2\binom{2k}kx^k$ for $n=0,1,2,...$. Those numbers $g_n=g_n(1)$ are closely related to Ap\'ery numbers and Franel numbers. In this paper we establish some fundamental congruences involving $g_n(x)$. For example, for any prime $p>5$ we have $$\sum_{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod{p^2}\quad{and}\quad\sum_{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p.$$ This is similar to Wolstenholme's classical congruences $$\sum_{k=1}^{p-1}\frac1k\equiv0\pmod{p^2}\quad{and}\quad\sum_{k=1}^{p-1}\frac{1}{k^2}\equiv0\pmod p$$ for any prime $p>3$.