# Two supercongruences related to multiple harmonic sums

## TL;DR

This paper establishes two new supercongruences involving truncated series of multiple harmonic sums for prime numbers and p-adic integers, extending the understanding of congruence properties in number theory.

## Contribution

It introduces novel supercongruences for specific truncated series involving multiple harmonic sums, expanding the theoretical framework of supercongruences in number theory.

## Key findings

- Proves two supercongruences for series involving multiple harmonic sums.
- Extends supercongruence results to series with p-adic parameters.
- Provides new tools for analyzing congruences in combinatorial and number-theoretic contexts.

## Abstract

Let $p$ be a prime and let $x$ be a $p$-adic integer. We provide two supercongruences for truncated series of the form $$\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}.$$