Convolution Identities on the Apostol-Hermite Base of Two Variables Polynomials

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 λ-Stirling numbers and two variables Hermite-Kampé de Fériet polynomials. © 2013 Foundation for Scientific Research and Technological Innovation.