q-Analogues of multiparameter non-central Stirling and generalized harmonic numbers B.S. El-Desouky a,⇑ , R.S. Gomaa a , Nenad P. Cakic ´ b a Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt b Department of Mathematics, Faculty of Electrical Engineering, University of Belgrade, P.O. Box 35-54, 11120 Belgrade, Serbia article info Keywords: Stirling numbers q-Stirling numbers Multiparameter non-central Stirling numbers Comtet numbers q-Analogue Harmonic numbers Generalized q-harmonic numbers abstract In this paper we derive q-analogues of the multiparameter non-central Stirling numbers of the first and second kind, introduced by El-Desouky. Moreover, recurrence relations, expli- cit formulas and a connection between these numbers and generalized q-harmonic num- bers are obtained. Furthermore, some important special cases and new combinatorial identities are given. Finally, algorithms of these numbers and matrix representation using Maple are derived. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction The multiparameter non-central Stirling numbers of first and second kind, respectively were introduced by El-Desouky [12] with ðtÞ n ¼ X n k¼0 sðn; k; aÞðt; aÞ k ; ð1:1Þ where a ¼ða 0 ; a 1 ; ... ; a n1 Þ, sð0; 0; aÞ¼ 1; sðn; k; aÞ¼ 0 for k > n, and ðt; aÞ n ¼ Q n1 i¼0 ðt  a i Þ, and ðt; aÞ n ¼ X n k¼0 Sðn; k; aÞðtÞ k ; ð1:2Þ where Sð0; 0; aÞ¼ 1; Sðn; k; aÞ¼ 0 for k > n. If a i ¼ i; i ¼ 0; 1; ... ; n  1, then ðt; aÞ n is reduced to ðtÞ n ¼ Q n1 i¼0 ðt  iÞ. Throughout this article we adopt the following notations from [3,4,14]. With 0 < q < 1; t a real number and n positive integer: ½t q ¼ 1  q t 1  q ; the q-number; ½t q ! ¼½t q ½t  1 q ½1 q ; ðt 2 N :¼f1; 2; 3; ...gÞ; the q-factorial of t; ð1:3Þ ½t n;q ¼½t q ½t  1 q ½t  n þ 1 q ; the falling factorial of q-number of order n; ð1:4Þ http://dx.doi.org/10.1016/j.amc.2014.01.026 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail address: b_desouky@yahoo.com (B.S. El-Desouky). Applied Mathematics and Computation 232 (2014) 132–143 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc