Some Families of Differential Equations Associated with the Hermite-Based Appell Polynomials and Other Classes of Hermite-Based Polynomials

Abstract

Recently, Khan {\itetal.} [S. Khan,G. Yasmin, R. Khan and N. A. M. Hassan, Hermite-based Appell polynomials:Properties and Applications, {\it J. Math. Anal.Appl.} {\bf 351} (2009), 756--764] defined the Hermite-based Appell polynomials by\begin{align*}\mathcal G(x,y,z;t)&:=A(t)\exp (xt+yt^{2}+zt^{3})\\ &\;=\sum_{n=0}^{\infty}\; _{H}A_{n}(x,y,z)\;\frac{t^{n}}{n!}\end{align*}and investigated their many interesting properties and characteristicsby using operational techniques combined with the principle of monomiality. Here, in this paper, we find the differential,integro-differential and partial differential equations for theHermite-based Appell polynomials via the factorization method. Furthermore,we derive the corresponding equations for the Hermite-based Bernoulli polynomials and the Hermite-based Euler polynomials. We also indicate how to deduce the corresponding results for the Hermite-based Genocchi polynomials from those involving the Hermite-based Euler polynomials.