STIRLING NUMBERS OF THE SECOND KIND

In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q; r; w)Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q; r; w)-Stirling numbers of the second kind and q-Bernoulli polynomials in w. 1. Introduction The Stirling numbers of the second kind S (n; k) de…ned by 1 X n=k S (n; k) t n! = (e 1) k! (n k; k 2 N0) (1.1) and their various generalizations have been studied by many mathematicians and physicists for a long time, see [2-8; 10-16]. For example, Broder [2] explored extensively the combinatorial and algebraic properties of the r-Stirling numbers, which is most cases generalize similar properties of the regular Stirling numbers. Carlitz [4] de…ned and studied an entirely di¤erent type of the generalization of the Stirling numbers, termed weighted Stirling numbers. Corcino et al. [6] established several properties for the q-analogue of the uni…ed generalizations of Stirling numbers including the vertical and horizontal recurrence relations, and the rational generating function. Guo et al. [7] derived an explicit formula for computing Bernoulli polynomials at non-negative integers in terms of r-Stirling numbers of the second kind. Guo et al. [8] reviewed some explicit formulas and set a novel explicit formula for Bernoulli and Genocchi numbers in terms of Stirling numbers of the second kind. Kim [10] considered the degenerate Stirling polynomials of the second kind which are derived from the generating function and proved some new identities for these polynomials. Kim et al. [11] provided several expressions, identities and properties 2000 Mathematics Subject Classi…cation. 11B68, 11B83, 81S40. Key words and phrases. q-Calculus; Stirling numbers of the second kind; Bernoulli polynomials and numbers; Generating function; Cauchy product. c 2017 Ilirias Research Institute, Prishtinë, Kosovë. Submitted August 1, 2007. Published January 2, 2008. The …rst author is thankful to The Scienti…c and Technological Research Council of Turkey (TUBITAK) for their support with the Ph.D. scholarship. The third author of this paper is also supported by the research fund of Hasan Kalyoncu University in 2017. 1 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 18 December 2017 doi:10.20944/preprints201712.0115.v1 © 2017 by the author(s). Distributed under a Creative Commons CC BY license. 2 U. DURAN, M. ACIKGOZ, S. ARACI about the extended degenerate Stirling numbers of the second kind and the extended degenerate Bell polynomials. Mahmudov [12] introduced and investigated a class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. In addition, he derived the q-analogues of well-known formulas including the q-analogue of the Srivastava-Pintér addition theorem. Mangontarum et al. [13] introduced the (q; r)-Whitney numbers of the …rst and second kinds in terms of the q-Boson operator and acquired some fundamental properties including recurrence formulas, orthogonality and inverse relations, and other interesting identities. They also obtained a q-analogue of the r-Stirling numbers of the …rst and second kinds. Mansour et al. [14] gave an one-stop reference on the research activities and known results of normal ordering and Stirling numbers and also de…ned associated generalized Stirling numbers as normal ordering coe¢ cients in analogy to the classical Stirling numbers. They discussed the Stirling numbers, closely related generalizations, and their role as normal ordering coe¢ cients in the Weyl algebra. Qi et al. [16] investigated a closed form for the Stirling polynomials in terms of the Stirling numbers of the …rst and second kinds by virtue of the Faá di Bruno formula and two identities for the Bell polynomial of the second kind. We use the following notations: N := f1; 2; 3; g and N0 := N [ f0g . As usual, R denotes the set of all real numbers. Based on the generating series (1.1), r-analogue of Stirling numbers of second kind is also given by the following generating function: 1 X n=k S (n+ r; k + r) t n! = (e 1) k! e (see [2; 5; 7; 12; 13]). (1.2) (n k and k 2 N0; r 2 N) We now give the de…nitions of some notations of q-calculus which can be found in [3,12,13]. The q-analogue of any real number x is de…ned by [x]q = q 1 q 1 (q 6= 1) . The q-derivative operator Dq of a function f is considered as (Dqf) (x) = f (x) f (qx) (1 q)x (q 6= 1 and x 6= 0) provided f (0) exists. For any two functions f (x) and g (x), the product rule of the q-derivative operator is given by Dq (f (x) g (x)) = g (qx)Dqf (x) + f (x)Dqg (x) . (1.3) The q-binomial coe¢ cients and q-factorial are de…ned, for positive integer n and k with n k, by n k q = [n]q! [n k]q! [k]q! and [n]q! = [n]q [n 1]q [n 2]q [1]q n 2 N; [0]q! = 1 : Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 18 December 2017 doi:10.20944/preprints201712.0115.v1