# 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.

## Key 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 $p>\max(n+1,3)$.