English

Two supercongruences related to multiple harmonic sums

Number Theory 2021-07-01 v1 Combinatorics

Abstract

Let pp be a prime and let xx be a pp-adic integer. We provide two supercongruences for truncated series of the form k=1p1(x)k(1)k1k1j1jrk1j1jr\mboxandk=1p1(x)k(1x)k(1)k21k1j1jrk1j12jr2.\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}}.

Keywords

Cite

@article{arxiv.1906.08741,
  title  = {Two supercongruences related to multiple harmonic sums},
  author = {Roberto Tauraso},
  journal= {arXiv preprint arXiv:1906.08741},
  year   = {2021}
}

Comments

This is a preliminary version. Comments are welcome

R2 v1 2026-06-23T09:59:13.313Z