Congruences of multiple sums involving invariant sequences under binomial transform

TL;DR

This paper establishes new congruences involving multiple sums of invariant sequences under binomial transforms, connecting them to Bernoulli polynomials modulo prime powers.

Contribution

It proves novel congruences for multiple sums involving invariant sequences under binomial transforms, extending known results to prime power moduli.

Findings

Derived congruences modulo p and p^2 for sums involving Bernoulli polynomials

Connected sums with binomial transform invariance to prime-related congruences

Extended the understanding of invariant sequences in modular arithmetic

Abstract

We will prove several congruences modulo a power of a prime such as $$ \sum_{0<k_1<...<k_{n}<p}\leg{p-k_{n}}{3} {(-1)^{k_{n}}\over k_1... k_{n}}\equiv {lll} -{2^{n+1}+2\over 6^{n+1}} p B_{p-n-1}({1\over 3}) &\pmod{p^2} &{if $n$ is odd} -{2^{n+1}+4\over n6^n} B_{p-n}({1\over 3}) &\pmod{p} &{if $n$ is even}. $$ where $n$ is a positive integer and $p$ is prime such that $p>\max(n+1,3)$.