Differential Equations Associated with Two Variable Degenerate Hermite Polynomials

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.

Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising from Differential Equations and Distribution of Their Zeros

“…We remember that the classical Stirling numbers of the first kind S 1 n, k ðÞ and the second kind S 2 n, k ðÞ are defined by the relations (see [6][7][8][9][10][11][12][13])…”

“…Recently, Hwang and Ryoo [10] proposed the 2-variable degenerate Hermite polynomials H n x, y, λ ðÞ by means of the generating function…”

“…1asλ approaches to 0, it is apparent that (10) descends to (7). Mathematicians have studied the differential equations arising from the generating functions of special numbers and polynomials (see [10][11][12][13][14]). Now, a new class of 2-variable modified degenerate Hermite polynomials are constructed based on the results so far.…”

Some Properties Involving 2-Variable Modified Partially Degenerate Hermite Polynomials Derived from Differential Equations and Distribution of Their Zeros

“…Motivated by their importance and potential for applications in certain problems in probability, combinatorics, number theory, differential equations, numerical analysis and other areas of mathematics and physics, several kinds of some special numbers and polynomials were recently studied by many authors (see [1,2,3,4,5,6,7]). Many mathematicians have studied in the area of the degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Genocchi polynomials, and degenerate tangent polynomials (see [ 6,7,8,9,10]).…”

Solving Some Differential Equations Arising in Electric Engineering Using Hermite Polynomials

“…Study of differential equations arising from the generating functions of Hermit Kampéde Fériet polynomials has been presented by Ryoo [9]. Hwang and Ryoo [10] investigate the structure and symmetry of zeros of the two variable degenerate Hermite equations. Singh [11] presented two numerical techniques such as Adomian decomposition and Haar wavelet methods for solving some oscillatory problems arising in several applications of science and engineering.…”