Differential Equations and Dynamical Systems

, Volume 22, Issue 3, pp 309–318

# Convolution Identities on the Apostol–Hermite Base of Two Variables Polynomials

• Yilmaz Simsek
Original Research

## Abstract

In this paper, we introduce a linear differential operator and investigate its fundamental properties. By means of this operator we derive convolution identities for Apostol–Hermite base two variables polynomials. These identities extend the Euler’s identities for the sums of product for the two variables Hermite base Apostol–Bernoulli and Apostol–Euler polynomials. Applying this differential operator to some specials functions, we obtain interesting identities and formulae involving the two variables Hermite base Apostol–Bernoulli and two variables Hermite base Apostol–Euler polynomials arising from the $$\lambda$$-Stirling numbers and two variables Hermite–Kampé de Fériet polynomials.

## Keywords

Convolution sums Apostol$$-$$Hermite polynomials Hermite$$-$$Kampé de Fériet $$\lambda$$-Stirling numbers

## Notes

### Acknowledgments

The first Author was supported by Laboratoire d’Analyse et probalités du département de mathématiques de l’université d’Evry, and the second Author was supported, by the Scientific Research Project Administration of Akdeniz University.

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