A Family of the r-Associated Stirling Numbers of the Second Kind and Generalized Bernoulli Polynomials

Paolo Emilio Ricci,Rekha Srivastava,Pierpaolo Natalini

doi:10.3390/axioms10030219

Paolo Emilio Ricci, Rekha Srivastava + Show 1 more

Open Access

https://doi.org/10.3390/axioms10030219

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Abstract

In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links with available literature on this subject are also pointed out. The extension to the bivariate case is discussed.

Highlights

Special functions and polynomials appear in the search for the explicit solutions of the most important problems in mathematical physics, electrodynamics, classical and modern physics, classical and quantum mechanics, and even in statistics and biological sciences
An important role is played by the hypergeometric functions which constitute an important class that unifies, through the introduction of appropriate parameters, most parts of special functions, including elliptic integrals, Beta functions, incomplete
The extension of the Pochhammer symbol allowed the introduction of many multivariate generalizations of hypergeometric functions

Summary

Introduction

Special functions and polynomials appear in the search for the explicit solutions of the most important problems in mathematical physics, electrodynamics, classical and modern physics, classical and quantum mechanics, and even in statistics and biological sciences. The Stirling numbers of the second kind are related to them through the equation: Bn n It seems that the basis of the generalizations of Bernoulli polynomials (and numbers) stands in the Mittag–Leffler function:. Which involves the r-associated Stirling numbers of the second kind This allows to represent the coefficients of such polynomials in function of the aforementioned r-associated numbers. To obtain this result, a general formula for the construction of the reciprocal of a power series is introduced which makes use of the Blissard problem and Bell’s polynomials. Denoting by ζn the coefficients of the series representing the ratio in brackets in Equation (27), the solution of the problem will be given by zn a0 b0 ζn

Generalized Bernoulli Polynomials

A Larger Class of Bernoulli Polynomials

Representation Formulas

The Generalized Bernoulli Numbers

Conclusions