Some families of differential equations associated with the Hermite-based Appell polynomials and other classes of Hermite-based polynomials

Recently, Khan { \it et al .} [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 characteristics by using operational techniques combined with the principle of monomiality . Here, in this paper, we find the differential, integro -differential and partial differential equations for the Hermite-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.