A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Ghulam Muhiuddin,Deena Al-Kadi,Waseem Ahmad Khan,Ugur Duran

doi:10.3390/sym13040648

Abstract

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

Highlights

We introduce the multivariable unified Lagrange–Hermite-based hypergeometric Bernoulli polynomials and investigate some of their properties
Where Gn ( x ) are called the Miller–Lee polynomials. Another example is the definition of higher-order hypergeometric Bernoulli–Hermite
We developed a point of view based on generating relations, exploited in the past, to study some aspects of the theory of special functions

Summary

Introduction

Special polynomials (like Bernoulli, Euler, Hermite, Laguerre, etc.) have great importance in applied mathematics, mathematical physics, quantum mechanics, engineering, and other fields of mathematics. For M, N ∈ N, and α ∈ C, Su and Komatsu [10] defined the hypergeometric Bernoulli polynomials B M,N,n ( x ) of order α by means of the following generating function: tn e xt. When x = 0, BM,N,n (0) := BM,N,n are the higher-order generalized hypergeometric Bernoulli numbers. Srivastava polynomials [2], are defined by means of the following generating function: r j =1 n =0. The multivariable (Erkus–Srivastava) polynomials Un;l1 ,··· ,lrr ( x1 , · · · , xr ) are defined by the following generating function [6]:. We introduce the multivariable unified Lagrange–Hermite-based hypergeometric Bernoulli polynomials and investigate some of their properties. We derive multifarious connected formulas involving the Miller–Lee polynomials, the Laguerre polynomials polynomials, the Lagrange Hermite–Miller–Lee polynomials

Lagrange–Hermite-Based Hypergeometric Bernoulli Polynomials

Some Connected Formulas

Conclusions