A Unified Family of Generalized $q$-Hermite Apostol Type Polynomials and its Applications

The intended objective of this paper is to introduce a new class of generalized $q$-Hermite based Apostol type polynomials by combining the $q$-Hermite polynomials and a unified family of $q$-Apostol-type polynomials. The generating function, series definition and several explicit representations for these polynomials are established. The $q$-Hermite-Apostol Bernoulli, $q$-Hermite-Apostol Euler and $q$-Hermite-Apostol Genocchi polynomials are studied as special members of this family and corresponding relations for these polynomials are obtained.

Keywords:

$q$-Hermite polynomials, Generalized $q$-Apostol type polynomials Generalized $q$-Hermite Apostol type polynomials, Explicit representation,

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