On Some Finite Sums with Factorials

TL;DR

This paper derives a summation formula involving factorials and explores its properties, including divisibility and connections to Kurepa's hypothesis, contributing new insights into factorial sums and their number-theoretic implications.

Contribution

It introduces a new summation formula involving factorials and polynomials, and investigates its divisibility properties and relation to Kurepa's hypothesis.

Findings

Derived a general summation formula involving factorials and polynomials.

Explored divisibility properties of the sum with respect to n.

Identified infinitely many equivalents to Kurepa's hypothesis.

Abstract

The summation formula $$ \sum^{n-1}_{i=0}\epsilon^i i! (i^k+u_k) = v_k+\epsilon^{n-1} n! A_{k-1}(n) $$ $(\epsilon=\pm 1; k=1,2,...; u_k, v_k\in \msbm\hbox{Z}; A_{k-1}$ is a polynomial) is derived and its various aspects are considered. In particular, divisibility with respect to $n$ is investigated. Infinitely many equivalents to Kurepa's hypothesis on the left factorial are found.