Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 951764, 9 pages
doi:10.1155/2010/951764
Research Article
A New Approach to q-Bernoulli
Numbers and q-Bernoulli Polynomials
Related to q-Bernstein Polynomials
Mehmet Ac¸ikg
¨
oz, Dilek Erdal, and Serkan Araci
Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey
Correspondence should be addressed to Mehmet Ac¸ikg
¨
oz, [email protected]
Received 24 November 2010; Accepted 27 December 2010
Academic Editor: Claudio Cuevas
Copyright q 2010 Mehmet Ac¸ikg
¨
oz 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 q-
Bernoulli 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.
|
< 2π, 1.1
and that B
n
B
n
0 are called the nth Bernoulli numbers.
2 Advances in Difference Equations
The recurrence formula f or the classical Bernoulli numbers B
n
is as follows,
B
0
1,
B 1
n
− B
n
0, if n>0 1.2
see 1, 3, 23.Theq-extension of the following recurrence f ormula for the Bernoulli numbers
is
B
0,q
1,q
qB 1
n
,
−x
q
−
1
q
x
x
q
,
xy
q
x
q
y
q
x
1.4
see 6.
x
q
k
k!
e
t1−x
q
∞
n0
B
k,n
x; q
t
n
n!
,t∈ C,k 0, 1, ,n
1.6
see 2, where lim
q → 1
F
k
x, t; qF
k
t, xtx
<p
−1/p−1
or |1 − q|
p
< 1sothatq
x
expx log q for |x|
p
≤ 1 see 7–19.
In this study, we present a new generating function related to the q-Bernoulli
numbers and q-Bernoulli polynomials and 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 q-Bernoulli 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
Advances in Difference Equations 3
which interpolates the q-Bernoulli polynomials at negative integers and are associated with
q-Bernstein polynomials.
2. New Approach to q-Bernoulli Numbers and Polynomials
Let N be the set of natural numbers and N
∗
N ∪{0}. For q ∈ C with |q| < 1, let us define the
q-Bernoulli polynomials B
n,q
x as follows,
D
q
t, x
t
e
t
− 1
e
xt
∞
n0
B
n
x
t
n
n!
,
|
t
|
< 2π,
2.2
where B
n
x are classical Bernoulli polynomials. In the special case x 0, B
n
n!
.
2.3
From 2.1 and 2.3,wenotethat
qD
q
t, 1
− D
q
t
qe
t
D
q
qt
− D
q
t
q
∞
n
l0
n
l
q
l
B
l,q
t
n
n!
−
∞
n0
B
n,q
t
n
n!
.
2.4
From 2.1 and 2.3, we can easily derive the following equation:
qD
q
0, if n>0.
2.6
Therefore, we obtain the following theorem.
4 Advances in Difference Equations
Theorem 2.1. For n ∈ N
∗
, one has
B
0,q
1,q
qB 1
n
− B
n,q
⎧
⎨
⎩
1, if n 0
0, if n>0.
2.7
with the usual convention of replacing B
i
and B
i,q
.
From 2.1, one notes that
D
n0
q
nx
B
n,q
t
n
n!
∞
n0
n
l0
n
l
q
lx
B
l,q
x
n−l
n−l
q
. 2.9
By 2.1, one sees that
D
q
t, x
∞
n0
−t
∞
m0
q
m
x m
n
q
t
n
n!
n
n!
.
2.10
By 2.1 and 2.10, one obtains the following theorem.
Theorem 2.3. For n ∈ N
∗
, one has
B
n,q
x
1
1 − q
n
n
l0
n
l
−1
l
x a
s
.
2.12
Advances in Difference Equations 5
By 2.12, one sees that, for s ∈ N,
∞
n0
B
n,q
x
t
n
n!
∞
n0
s
n
q
s−1
q
s−1
a0
q
a
B
n,q
s
x a
s
.
2.14
In 2.9, substitute 1 − x instead of x,oneobtains
B
n,q
1 − x
n
v0
n
v
B
x
−v
q
∞
m0
n
v0
B
v,n
x; q
v m − 1
m
q
v
1 − q
m
x
.
2.16
In 2.16, substitute m − v instead of k, and putting the result in 2.15, one has the following
theorem.
Theorem 2.5. For n ∈ N
∗
and |q| < 1, one has
B
n,q
x
∞
m,y0
n
v0
v
j0
v m − 1
m
v
j
6 Advances in Difference Equations
Let χ be Dirichlet’s character with s ∈ N. Then, one defines the generalized q-Bernoulli
polynomials attached to χ as follows,
D
q,χ
t, x
−t
∞
d0
χ
d
q
d
e
dx
q
t
∞
n0
B
n,χ,q
x
q
t
∞
n0
B
n,χ,q
t
n
n!
.
2.19
By 2.1 and 2.18, one sees that
D
q,χ
t, x
s−1
a0
q
a
χ
a
D
B
n,q
s
x a
s
t
n
n!
.
2.20
Therefore, one obtains the following theorem.
Theorem 2.6. For n ∈ N
∗
and s ∈ N, one has
B
n,χ,q
x
s
n
q
s−1
q
x
t
∞
n0
n
d0
n
d
q
dx
x
n−d
q
B
d,χ,q
t
n
n!
. 2.22
Γ
s
∞
0
D
q
−t, x
t
s−2
dt
∞
n0
q
n
x n
s
q
,
2.24
for s ∈ C,andx
/
0, −1, −2,
1 − n, x
B
n,q
x
, for n ∈ N
∗
.
2.26
By the same method, one can also obtain the following equations:
1
Γ
s
∞
0
D
q,χ
−t, x
t
s−2
dt
∞
n
n x
s
q
,
2.28
where x
/
0, −1, −2, Notethatl
q
s, x | χ is also a holomorphic function in the whole
complex s-plane. From the Laurent series and the Cauchy residue theorem, one can also
derive the following equation:
−nl
q
1 − n, x | χ
B
n,χ,q
x
. 2.29
In 2.23, substitute 1 − x instead of x,oneobtains
B
n,χ,q
v
q
1 − x
n−v
q
B
v,χ,q
· q
v1−x
x
−v
q
∞
m0
n
v0
B
v,n
x; q
v m − 1
∞
m,y0
n
v0
v
j0
v m − 1
m
v
j
−1
m−vj
m − v
!q
vj
y!
× S
y, m − v
s
q
.
2.32
From 2.32, one has
H
q
s, a | F
q
a
F
s
q
ζ
∗
q
F
s,
a
F
, 2.33
where ζ
∗
a
F
−
nH
q
1 − n, a | F
q
a
F
n−1
q
.
2.35
References
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¨
oz and S. Aracı, “The relations between Bernoulli, Bernstein and Euler polynomials,” in
Proceedings of the 8th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM
’10), AIP, Rhodes, Greece, March 2010.
2 M. Ac¸ıkg
¨
oz and Y. S¸ims¸ ek, “A new generating function of q-Bernstein-type polynomials and their
interpolation function,” Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010.
3 M. Ac¸ıkg
, vol. 2010, Article ID 706483, 12 pages, 2010.
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.org/abs/1008.4547.
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integrals,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 179430, 9 pages, 2010.
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and q-Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 20, pp. 335–341, 2010.
19 T. Kim, “Analytic continuation of multiple q-zeta functions and their values at negative integers,”
Russian Journal of Mathematical Physics, vol. 11, no. 1, pp. 71–76, 2004.
20 T. Kim, “Note on the Euler q-zeta functions,” Journal of Number Theory, vol. 129, no. 7, pp. 1798–1804,
2009.
21 T. Kim, “Barnes-type multiple q-zeta functions and q-Euler polynomials,” Journal of Physics A, vol. 43,
no. 25, Article ID 255201, 11 pages, 2010.
22 T. Kim, S H. Rim, J W. Son, and L C. Jang, “On the values of q-analogue of zeta and L-functions. I,”
Algebra Colloquium, vol. 9, no. 2, pp. 233–240, 2002.
23 H. Rademacher, Topics in Analytic Number Theory, Springer, New York, NY, USA, 1973.
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