Some array type polynomials associated with special numbers and polynomials Abdelmejid Bayad a , Yilmaz Simsek b , H.M. Srivastava c,⇑ a Laboratoire Analyse et Probabilités, Université d’Evry Val d’Essonne, 23 Boulevard de France, F-91037 Evry, France b Department of Mathematics, Faculty of Science, University of Akdeniz, TR-07058 Antalya, Turkey c Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada article info Keywords: Bernoulli polynomials and Bernoulli numbers Apostol–Bernoulli polynomials and Apostol–Bernoulli numbers k-Bell numbers and k-Bell polynomials k-Array polynomials k-Stirling numbers of the second kind abstract The main objective in this paper is first to establish new identities for the k-Stirling type numbers of the second kind, the k-array type polynomials, the Apostol–Bernoulli polyno- mials and the Apostol–Bernoulli numbers. We then construct a k-delta operator and inves- tigate various generating functions for the k-Bell type numbers and for some new polynomials associated with the k-array type polynomials. We also derive several other identities and relations for these polynomials and numbers. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction, notations and preliminaries 1.1. Notations and motivation Throughout this paper, we use the following standard notations: N ¼f1; 2; 3; ...g; N 0 ¼f0; 1; 2; 3; ...g¼ N [f0g and Z  ¼ f1; 2; 3; ...g: As usual, Z denotes the set of integers, R is the set of real numbers, C is the set of complex numbers and Z p is the set of p-adic integers. The principal value ln z is the logarithm whose imaginary part lies in the interval ðp; p. Furthermore, in this paper, we use the following notational conventions: 0 n ¼ 1 ðn ¼ 0Þ 0 ðn 2 NÞ;  k 0  ¼ 1 and k n  ¼ kðk  1Þðk  n þ 1Þ n! ðn 2 N; k 2 CÞ and k fg 0 ¼ 1 and k fg n ¼ Y n1 j¼0 ðk  jÞ ðn 2 N; k 2 CÞ: http://dx.doi.org/10.1016/j.amc.2014.06.086 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: abayad@maths.univ-evry.fr (A. Bayad), ysimsek@akdeniz.edu.tr (Y. Simsek), harimsri@math.uvic.ca (H.M. Srivastava). Applied Mathematics and Computation 244 (2014) 149–157 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc