# On poly-Bell numbers and polynomials

## TL;DR

This paper introduces a new family of poly-Bell numbers and polynomials related to Bell numbers via the confluent hypergeometric function, exploring their properties and applications in combinatorics and analysis.

## Contribution

It constructs and analyzes poly-Bell numbers and polynomials, providing explicit formulas, generating functions, and integral representations, expanding the theory of Bell-related special functions.

## Key findings

- Derived explicit formulas and generating functions for poly-Bell numbers.
- Established recurrence relations and integral representations.
- Connected poly-Bell numbers to combinatorial sums and special functions.

## Abstract

This paper aims to construct a new family of numbers and polynomials which are related to the Bell numbers and polynomials by means of the confluent hypergeometric function. We give various properties of these numbers and polynomials (generating functions, explicit formulas, integral representations, recurrence relations, probabilistic representation,...). We also derive some combinatorial sums including the generalized Bernoulli polynomials, lower incomplete gamma function, generalized Bell polynomials. Finally, by applying Cauchy formula for repeated integration, we introduce poly-Bell numbers and polynomials.