Research Article
Received 15 October 2010 Published online 19 April 2012 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.1580
MOS subject classification: 11B68; 11B39; 11B73; 05A15; 14F10; 12D10; 26C05; 26C10; 30B40; 30C15
q-Bernstein polynomials related to
q-Frobenius–Euler polynomials, l -functions,
and q-Stirling numbers
Yilmaz Simsek
a
*
†
, Abdelmejid Bayad
b
and V. Lokesha
c
Communicated by W. Sprößig
The aim of this paper was to derive new identities and relations associated with the q-Bernstein polynomials, q-Frobenius–
Euler polynomials, l-functions, and q-Stirling numbers of the second kind. We also give some applications related to theses
polynomials and numbers. Copyright © 2012 John Wiley & Sons, Ltd.
Keywords: q-Bernstein polynomials; q-Frobenius–Euler polynomials; l-functions; q-Stirling numbers of the second kind
1. Introduction
Throughout this paper, we make use of the following notations:
C denotes the set of complex numbers and
N
0
D N[f0g , N Df1, 2, 3, g
denotes the set of non-negative integers.
For q 2 C (jqj < 1), the q-number Œa, q and the q-number factorial Œa, qŠ are defined by (see, for details, [1])
Œ0, q D 0, and Œa, q D
1 q
a
1 q
,
where q ¤ 1; a 2 C,
Œ0, qŠ D 1 and Œk, qŠ D Œ1, qŠ Œ2, qŠ Œ3, qŠ Œk, qŠ,
where k 2 N, respectively. One has the following limit cases:
lim
q!1
Œa, q D a and lim
q!1
Œn, qŠ D nŠ
cf. [2–16].
For k, l, m, n 2 N,
n
k
D
nŠ
kŠ.n k/Š
, n k
and
a
Department of Mathematics, Faculty of Science, University of Akdeniz , TR-07058 Antalya, Turkey
b
Département de mathématiques, Université d’Evry Val d’Essone , Bd. F. Mitterrand, 91025 Evry Cedex, France
c
Department of Mathematics, Acharya Institute of Technology , Soldevanahalli, Bangalore-90, India
*Correspondence to: Yilmaz Simsek, Department of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya, Turkey.
†
E-mail: ysimsek@akdeniz.edu.tr
Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012, 35 877–884
877