Abstract
In this paper, we introduce a unified family of Hermite-based Apostol-Bernoulli, Euler and Genocchi polynomials. We obtain some symmetry identities between these polynomials and the generalized sum of integer powers. We give explicit closed-form formulae for this unified family. Furthermore, we prove a finite series relation between this unification and 3d-Hermite polynomials. MSC:11B68, 33C05.
Highlights
Khan et al [ ] introduced the Hermite-based Appell polynomials via the generating functionG(x, y, z; t) = A(t) exp(Mt), where ∂ M = x +y ∂x z ∂x is the multiplicative operator of the -variable Hermite polynomials, which are defined by exp xt + yt + zt ∞Hn( )(x, y, z) tn n! ( . )
In Section, we introduce the unification of the Hermite-based generalized ApostolBernoulli, Euler and Genocchi polynomials H Pn(α,β) (x, y, z; k, a, b) and give summation formulas for this unification
The Hermite-based generalized Apostol-Bernoulli polynomials are defined by t λet
Summary
2 Hermite-based generalized Apostol-Bernoulli, Euler and Genocchi polynomials The Hermite-based generalized Apostol-Bernoulli polynomials are defined by t λet – The Hermite-based generalized Apostol-Euler polynomials are defined by
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