Differential Equations Associated with Two Variable Degenerate Hermite Polynomials

Kyung-Won Hwang,Cheon Seoung Ryoo

doi:10.3390/math8020228

Abstract

In this paper, we introduce the two variable degenerate Hermite polynomials and obtain some new symmetric identities for two variable degenerate Hermite polynomials. In order to give explicit identities for two variable degenerate Hermite polynomials, differential equations arising from the generating functions of degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the two variable degenerate Hermite equations.

Highlights

The classical Hermite numbers Hn and Hermite polynomials Hn ( x ) are usually defined by the generating functions: ∞tn e2xt−t = ∑ Hn ( x ) n! n =0 and e−t = ∑ Hn n! . n =0Clearly, Hn = Hn (0).It can be seen that these numbers and polynomials play an important role in various areas of mathematics and physics, including numerical theory, combinations, special functions, and differential equations
In mathematics and physics, the Hermite polynomials are a classical orthogonal polynomial sequence. They appears as the Edgeworth series; in combinatorics, they arise in the umbral calculus as an example of an Appell sequence; in numerical analysis, they play a role in Gaussian quadrature; and in physics, they give rise to the eigenstates of the quantum harmonic oscillator
We constructed differential equations arising from the generating function of the two variable degenerate Hermite polynomials

Summary

Introduction

The classical Hermite numbers Hn and Hermite polynomials Hn ( x ) are usually defined by the generating functions:. Mathematics 2020, 8, 228 special polynomials of mathematical physics and their generalization have been proposed by physical problems As another application of the Hermite differential equation for Hn ( x, y), we recall that the two variable Hermite polynomials Hn ( x, y) defined by the generating function (see [2]):. Motivated by their potential and importance for applications in certain problems in probability, combinatorics, number theory, differential equations, numerical analysis, and other fields of mathematics and physics, several kinds of some special polynomials and numbers were recently studied by many authors (see [1,2,3,4,5,6,7]).

Basic Properties for the Two Variable Degenerate Hermite Polynomials

Symmetric Identities for the Two Variable Degenerate Hermite Polynomials

Zeros of the Two Variable Degenerate Hermite Polynomials

Conclusions and Future Directions