Generalized Bell polynomials

TL;DR

This paper introduces generalized Bell polynomials linked to a sequence of parameters, explores their zero properties, and establishes their equivalence to Laguerre multiple polynomials of the first kind.

Contribution

It defines a new class of generalized Bell polynomials, analyzes their zero distribution, and connects them to Laguerre multiple polynomials, expanding understanding of polynomial families.

Findings

Zeros are simple, real, and non-positive when parameters are non-negative.

Zeros of successive polynomials interlace.

Generalized Bell polynomials are equivalent to Laguerre multiple polynomials of the first kind.

Abstract

In this paper, generalized Bell polynomials $(\Be_n^\phi)_n$ associated to a sequence of real numbers $\phi=(\phi_i)_{i=1}^\infty$ are introduced. Bell polynomials correspond to $\phi_i=0$, $i\ge 1$. We prove that when $\phi_i\ge 0$, $i\ge 1$: (a) the zeros of the generalized Bell polynomial $\Be_n^\phi$ are simple, real and non positive; (b) the zeros of $\Be_{n+1}^\phi$ interlace the zeros of $\Be_n^\phi$; (c) the zeros are decreasing functions of the parameters $\phi_i$. We find a hypergeometric representation for the generalized Bell polynomials. As a consequence, it is proved that the class of all generalized Bell polynomials is actually the same class as that of all Laguerre multiple polynomials of the first kind.