# Some Families of Super Congruences Involving Alternating Multiple Harmonic Sums

## TL;DR

This paper investigates super congruences involving alternating multiple harmonic sums, extending previous results to broader families and providing new insights into their properties modulo prime powers.

## Contribution

It introduces new families of super congruences involving alternating sums, generalizing earlier specific cases for n=4 and 5.

## Key findings

- Extended super congruences for alternating sums modulo prime powers.
- General formulas that encompass previous special cases.
- Deeper understanding of the structure of alternating multiple harmonic sums.

## Abstract

Let $p$ be a prime. In this short note we study some families of super congruences involving the following alternating sums \begin{equation*} \sum_{\substack{j_1+j_2+\cdots+j_n=2 p^r p\nmid j_1 j_2 \cdots j_n}} \frac{(-1)^{j_1+\cdots+j_b}}{j_1\cdots j_n} \pmod{p^r}, \end{equation*} which extend similar statements proved by Shen and Cai who treated the cases when $n=4,5$.