Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials
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Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials Mehmet Ali Özarslan* *
Correspondence: [email protected] Department of Mathematics, Faculty of Arts of Sciences, Eastern Mediterranean University, Gazimagusa, North Cyprus, Mersin 10, Turkey
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: Primary 11B68; secondary 33C05 Keywords: Hermite-based Apostol-Bernoulli polynomials; Hermite-based Apostol-Euler polynomials; Hermite-based Apostol-Genocchi polynomials; generalized sum of integer powers; generalized sum of alternative integer powers
1 Introduction Recently, Khan et al. [] introduced the Hermite-based Appell polynomials via the generating function G (x, y, z; t) = A(t) exp(Mt), where
M = x + y
∂ ∂ + z ∂x ∂x
is the multiplicative operator of the -variable Hermite polynomials, which are defined by ∞ tn exp xt + yt + zt = Hn() (x, y, z) n! n=
(.)
and A(t) =
∞
an t n ,
a = .
n=
By using the Berry decoupling identity, / (( –m )A/ +A) B
eA+B = em
e
e ,
[A, B] = mA/
© 2013 Özarslan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Özarslan Advances in Difference Equations 2013, 2013:116 http://www.advancesindifferenceequations.com/content/2013/1/116
Page 2 of 13
they obtained the generating function of the Hermite-based Appell polynomials H An (x, y, z) as ∞ tn G (x, y, z; t) = A(t) exp xt + yt + zt = H An (x, y, z) . n! n=
Letting A(t) =
t , et –
they defined Hermite-Bernoulli polynomials H Bn (x, y, z) by
∞ tn t = exp xt + yt , + zt B (x, y, z) n H et – n! n=
For A(t) =
, et +
they defined Hermite-Euler polynomials H En (x, y, z) by
∞ tn = exp xt + yt + zt H En (x, y, z) , t e + n! n=
and for A(t) =
|t| < π.
t , et +
|t| < π
they defined Hermite-Genocchi polynomials H Gn (x, y, z) by
∞ tn t = exp xt + yt , + zt G (x, y, z) n H et + n! n=
|t| < π.
Recently, the author considered the following unification of the Apostol-Bernoulli, Euler and Genocchi polynomials (α) fa,b (x; t; k, β) :=
–k t k β b et – ab
α ext =
k ∈ N ; a, b ∈ R\{}; α, β ∈ C
∞
(α) Pn,β (x; k, a, b)
n=
tn n!
and obtained the explicit representation of this unified family, in terms of Gaussian hypergeometric function. Some symmetry identities and multiplication formula are also given () (x, y, z; k, a, b) was investigated in []. in []. Note that the family of polynomials Pn,β We
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