Differential Equations Arising from the 3-Variable Hermite Polynomials and Computation of Their Zeros

Abstract

In this paper, we study differential equations arising from the generating functions of the 3-variable Hermite polynomials. We give explicit identities for the 3-variable Hermite polynomials. Finally, we investigate the zeros of the 3-variable Hermite polynomials by using computer.

Computational formulas and identities for new classes of Hermite‐based Milne–Thomson type polynomials: Analysis of generating functions with Euler's formula

“…where [x] denotes the largest integer ≤x (cf. previous literature 4,7,11,15,21,24 ). The generalized Hermite-Kampè de Fèriet polynomials H n ( ⃗ u, r ) are defined by means of the following generating function:…”

Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations

“…Differential equations arising from the generating functions of special polynomials are studied by many authors to give explicit identities for special polynomials (see [4][5][6][7][8]). In this section, we study differential equations arising from the generating functions of (r, β)-Bell polynomials.…”

“…Recently, many mathematicians have studied the differential equations arising from the generating functions of special polynomials (see [4][5][6][7][8]). Inspired by their work, we give a differential equations by generation of (r, β)-Bell polynomials G n (x, r, β) as follows.…”