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

H. M. Srivastava, M. A. Özarslan, Banu Yılmaz

Recently, Khan et al. [S. Khan, G. Yasmin, R. Khan and N. A. M. Hassan, Hermite-based Appell polynomials: Properties and Applications, J. Math. Anal. Appl. 351 (2009), 756--764] defined the Hermite-based Appell polynomials by \[ \Cal G(x,y,z;t):=A(t)\cdot\exp(xt+yt^2+zt^3)=\sum^\infty_{n=0}{_HA_n(x,y,z)}\frac{t^n}{n!} \] 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.