Some properties of degenerate Hermite Appell polynomials in three variables

Abstract

This study presents Δh-Hermite Appell polynomials in three variables which include both discrete and degenerate cases. We investigate some properties of them such as explicit representation, determinantal form, recurrence relation, lowering operators (LO), raising operators (RO), difference equation (DE), integro-difference equation (IDE) and partial difference equation (PDE). We also obtain the explicit expression in terms of the Stirling numbers of the first kind. Moreover, we introduce 3D-Δh-Hermite Charlier polynomials, 3D-Δh-Hermite Carlitz Bernoulli polynomials, 3D-Δh-Hermite Carlitz Euler polynomials, 3D-Δh-Hermite Boole polynomials as special cases of Δh-Hermite Appell polynomials. Furthermore, we derive the explicit representation, determinantal form, recurrence relation, LO, RO and DE for these special cases. Finally, we introduce new approximating operators based on h-Hermite polynomials in three variables and examine the weighted Korovkin theorem. The error of approximation is also calculated in terms of the modulus of continuity and Peetre’s K-functional.