On the supercongruences involving harmonic numbers of order 2

TL;DR

This paper proves new supercongruences involving harmonic numbers of order two, confirming conjectures and providing explicit modulo p^3 results for specific sums with binomial coefficients.

Contribution

It establishes explicit supercongruences for sums involving harmonic numbers of order two and binomial coefficients, confirming conjectures by Z.-W. Sun.

Findings

Complete determination of specific sums modulo p^3

Confirmation of two conjectured congruences by Sun

Results valid for primes greater than 5

Abstract

We prove several supercongruences involving the harmonic number of order two $H_n^{(2)}:=\sum_{k=1}^n1/k^2$. For example, if $p>5$ is prime and $\alpha$ is $p$-integral, then we can completely determine $$ \sum_{k=0}^{p-1}\frac{H_k^{(2)}}{k}\cdot\binom{\alpha}{k}\binom{-1-\alpha}{k}\quad\text{and}\quad \sum_{k=0}^{\frac{p-1}{2}}\frac{H_k^{(2)}}{k}\cdot\binom{\alpha}{k}\binom{-1-\alpha}{k} $$ modulo $p^3$. In particular, by setting $\alpha=-1/2$, we confirm two conjectured congruences of Z.-W. Sun.