analog of Apostol type polynomials of order

Motivated by Kurt’s work [Filomat 30 (4) 921-927, 2016], we …rst consider a class of a new generating function for (p; q)-analog of Apostol type polynomials of order including Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order . By making use of their generating function, we derive some useful identities. We also introduce (p; q)-analog of Stirling numbers of second kind of order v by which we construct a relation including aforementioned polynomials. 2010 Mathematics Subject Classi…cationPrimary 05A30; Secondary 11B68, 11B73. Keywords and Phrases(p; q)-calculus; Apostol-Bernoulli polynomials; Apostol-Euler polynomials; ApostolGenocchi polynomials; Stirling numbers of second kind; Generating function; Cauchy product. 1. Introduction During the last three decades, applications of quantum calculus based on q-numbers have been studied and investigated succesfully, densely and considerably (see [4; 8; 10; 15; 16; 19-21; 27; 30; 31]). In conjunction with the motivation and inspiration of these applications, with the introduction of the (p; q)-number, many mathematicians and physicists have extensively developed the theory of post quantum calculus based on (p; q)-numbers along the traditional lines of classical and quantum calculus. Certainly, these (p; q)-numbers cannot be derived only switching q by q=p in q-numbers. Conversely, (p; q)-numbers are native generalizations of q-numbers, since q-numbers may be obtained when p = 1 in the de…nition of (p; q)-numbers (see [10]). In recent years, Corcino [5] studied on the (p; q)-extension of the binomial coe¢ cients and also derived some properties parallel to those of the ordinary and q-binomial coe¢ cients, comprised horizontal generating function, the triangular, vertical, and the horizontal recurrence relations, and the inverse and the orthogonality relationships. Duran et al.[6] considered (p; q)-analogs of Bernoulli polynomials, Euler polynomials and Genocchi polynomials and acquired the (p; q)-analogues of known earlier formulae. Duran and Acikgoz [7] gave (p; q)-analogue of the Apostol-Bernoulli, Euler and Genocchi polynomials and derived their some properties. Gupta [10] proposed the (p; q)-variant of the Baskakov-Kantorovich operators by means of (p; q)-integrals and also analyzed some approximation properties of them. Milovanovíc et al. [22] introduced a generalization of Beta functions under the (p; q)-calculus and committed the integral modi…cation of the generalized Bernstien polynomials. Sadjang [26] satis…ed some properties of the (p; q)-derivatives and the (p; q)-integrations. As an application, he presented two (p; q)-Taylor formulas for polynomials and derived the fundamental theorem of (p; q)-calculus. The (p; q)-number is de…ned as [n]p;q = p q p q (p 6= q) . Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 21 December 2017 doi:10.20944/preprints201712.0152.v1 © 2017 by the author(s). Distributed under a Creative Commons CC BY license. 2 U. Duran, M. Acikgoz and S. Araci Notice that [n]1;q := [n]q called as q-number in q-calculus. The (p; q)-derivative operator given as Dp;q;xf (x) := Dp;qf (x) = f (px) f (qx) (p q)x (x 6= 0) with (Dp;qf) (0) = f 0 (0) (1.1) is a lineer operator and satis…es the following properties Dp;q (f (x) g (x)) = f (px)Dp;qg (x) + g (qx)Dp;qf (x) (1.2) and Dp;q f (x) g (x) = g (px)Dp;qf (x) f (px)Dp;qg (x) g (px) g (qx) . (1.3) The (p; q)-power basis is de…ned by (x+ a) n p;q := (x+ a)(px+ aq) (p x+ aq )(p x+ aq ), if n 1, 1, if n = 0, or equivalently by = n P