Abstract
Recently, Choi et al. (2008) have studied theq-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of ordernand multiple Hurwitz zeta function. In this paper, we define Apostol's typeq-Euler numbersEn,q,ξandq-Euler polynomialsEn,q,ξ(x). We obtain the generating functions ofEn,q,ξandEn,q,ξ(x), respectively. We also have the distribution relation for Apostol's typeq-Euler polynomials. Finally, we obtainq-zeta function associated with Apostol's typeq-Euler numbers and Hurwitz's typeq-zeta function associated with Apostol's typeq-Euler polynomials for negative integers.
Highlights
1.7 and these numbers are interpolated by the Euler zeta function which is defined as ζE s
Q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials and numbers have been studied by many authors with great interest
Choi et al 16 have studied some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n and multiple Hurwitz zeta function
Summary
Choi et al 2008 have studied the q-extensions of the Apostol-Bernoulli and the ApostolEuler polynomials of order n and multiple Hurwitz zeta function. We define Apostol’s type q-Euler numbers En,q,ξ and q-Euler polynomials En,q,ξ x. We obtain the generating functions of En,q,ξ and En,q,ξ x , respectively. We have the distribution relation for Apostol’s type q-Euler polynomials. We obtain q-zeta function associated with Apostol’s type q-Euler numbers and Hurwitz’s type q-zeta function associated with Apostol’s type q-Euler polynomials for negative integers.
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