A New Approach to -Bernoulli Numbers and -Bernoulli Polynomials Related to -Bernstein Polynomials
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Research Article A New Approach to q-Bernoulli Numbers and q-Bernoulli Polynomials Related to q-Bernstein Polynomials ¨ Dilek Erdal, and Serkan Araci Mehmet Ac¸ikgoz, Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey Correspondence should be addressed to Mehmet Ac¸ikgoz, ¨ [email protected] Received 24 November 2010; Accepted 27 December 2010 Academic Editor: Claudio Cuevas Copyright q 2010 Mehmet Ac¸ikgoz ¨ et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We present a new generating function related to the q-Bernoulli numbers and q-Bernoulli polynomials. We give a new construction of these numbers and polynomials related to the second-kind Stirling numbers and q-Bernstein polynomials. We also consider the generalized qBernoulli polynomials attached to Dirichlet’s character χ and have their generating function. We obtain distribution relations for the q-Bernoulli polynomials and have some identities involving q-Bernoulli numbers and polynomials related to the second kind Stirling numbers and q-Bernstein polynomials. Finally, we derive the q-extensions of zeta functions from the Mellin transformation of this generating function which interpolates the q-Bernoulli polynomials at negative integers and is associated with q-Bernstein polynomials.
1. Introduction, Definitions, and Notations Let C be the complex number field. We assume that q ∈ C with |q| < 1 and that the q-number is defined by xq qx − 1/q − 1 in this paper. Many mathematicians have studied q-Bernoulli, q-Euler polynomials, and related topics see 1–23. It is known that the Bernoulli polynomials are defined by ∞ t tn xt , e B x n n! et − 1 n0
for |t| < 2π,
and that Bn Bn 0 are called the nth Bernoulli numbers.
1.1
2
Advances in Difference Equations The recurrence formula for the classical Bernoulli numbers Bn is as follows, B 1n − Bn 0,
B0 1,
if n > 0
1.2
see 1, 3, 23. The q-extension of the following recurrence formula for the Bernoulli numbers is ⎧ ⎨1, n q qB 1 − Bn,q ⎩0,
B0,q 1,
if n 1, if n > 1,
1.3
with the usual convention of replacing Bn by Bn,q see 5, 7, 14. Now, by introducing the following well-known identities x y q xq qx y q ,
−xq −
1 xq , qx
xy q xq y qx
1.4
see 6. The generating functions of the second kind Stirling numbers and q-Bernstein polynomials, respectively, can be defined as follows, k ∞ et − 1 tn Sn, k , k! n! n0
Fk x, t; q
k
txq k!
et1−xq
∞
tn , Bk,n x; q n! n0
1.5
t ∈ C, k 0, 1, . . . , n
1.6
see 2, where limq → 1 Fk x, t; q Fk t, x txk /k!et1−x see 4. Throughout this paper, Z, Q, Zp , Qp , and Cp will respectively denote the ring of rational integers, the field of rational numbers, the ring p-adic rational integers, the field of
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