# Identities and congruences involving the Fubini polynomials

## TL;DR

This paper explores the umbral representation of Fubini polynomials to derive properties and congruences, including a key result involving prime numbers and polynomial congruences.

## Contribution

It introduces new congruences involving Fubini polynomials and their umbral representations, extending understanding of their algebraic properties.

## Key findings

- Proves that (f(F_x))^p ≡ f(F_x) mod p for prime p.
- Derives several novel congruences involving Fubini polynomials.
- Provides insights into the algebraic structure of Fubini polynomials.

## Abstract

In this paper, we investigate the umbral representation of the Fubini polynomials $F_{x}^{n}:=F_{n}(x)$ to derive some properties involving these polynomials. For any prime number $p$ and any polynomial $f$ with integer coefficients, we show $(f(F_{x}))^{p}\equiv f(F_{x})$ and we give other curious congruences.