Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 743295, 10 pages
doi:10.1155/2008/743295
Research Article
q-Bernoulli Numbers Associated with
q-Stirling Numbers
Taekyun Kim
Division of General Education-Mathematics, Kwangwoon University, Seoul 139704, South Korea
Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr
Received 13 December 2007; Accepted 29 January 2008
Recommended by Panayiotis D. Siafarikas
We consider Carlitz q-Bernoulli numbers and q-Stirling numbers of the first and the second kinds. From the properties of q-Stirling numbers, we derive many interesting
n formun
las associated with Carlitz q-Bernoulli numbers. Finally, we will prove βn,q m0
km 1/
k
nm−k n−m
i0 idi s
q
k,
m−1
m
1/m
1
,
where
β
are
called
Carlitz
q1 − q
1,q
q
d0 ···dk n−k
n,q
Bernoulli numbers.
Copyright q 2008 Taekyun Kim. 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.
1. Introduction
Let p be a fixed prime number. Throughout this paper, Zp , Qp , C, and Cp will, respectively,
denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex
number field, and the completion of algebraic closure of Qp . For d a fixed positive integer with
p, d 1, let
X Xd lim Z dpN Z,
X∗ 0<a<dp
a,p1
←−
N
X1 Zp ,
a dpZp ,
1.1
a dpN Zp x ∈ X | x ≡ a mod dpN ,
where a ∈ Z lies in 0 ≤ a < dpN , see 1–21. The p-adic absolute value in Cp is normalized so that |p|p 1/p. When one talks about q-extension, q is variously considered as an
indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp . If q ∈ Cp , then we
2
Advances in Difference Equations
assume |q − 1|p < p−1/p−1 , so that qx expx log q for |x|p ≤ 1. We use the notation
xq x : q 1 − qx /1 − q. For f ∈ C1 Zp {f | f ∈ CZp }, let us start with the
expressions
1
qj fj fjμq j pN Zp
N
p q 0≤j<pN
0≤j<pN
1.2
see 6, 8, representing q-analogue of Riemann sums for f. The p-adic q-integral of a function
f ∈ C1 Zp is defined by
fxdμq x X
Zp
fxdμq x lim N→∞
1
p
N
N
−1
p
fxqx
see 8, 22, 23. For f ∈ C1 Zp , it is easy to see that
fxdμq x ≤ p
f
1
Zp
1.3
q x0
1.4
p
see 6–14, where f
1 sup{|f0|p , supx/y |fx − fy/x − y|p }. If fn → f in C1 Zp ,
namely, fn − f
1 → 0, then
fn xdμq x −→
fxdμq x
1.5
Zp
Zp
x
see 6–10. The q-analogue of binomial coefficient was known as n q xq x − 1q · · · x
n
− n 1q /nq !, where nq ! i1 iq see 1, 5, 6, 10, 11. From this definition, we derive
x
x
x1
x
x
n
x−n
q
q
1.6
n−1 q
n q
n q
n q
n−1 q
n1 x
x
n1−
2
cf. 6, 10. Thus, we have Zp n q dμq x −1n /n 1q q
. If fx k≥0 ak,q k q
is the q-analogue of Mahler series of strictly differentiable function f, then we see that
−1k k1− k1
2
.
1.7
fxdμq x ak,q
q
k 1q
Zp
k≥0
Carlitz q-Bernoulli numbers βk,q βk q can be determined inductively by
⎧
⎨1 if k 1,
β0,q 1,
qqβ 1k − βk,q ⎩0 if k > 1,
1.8
with the usual convention of replacing βi by βi,q see 2–4. In this paper, we study the qStirling numbers of the first and the second kinds. From these q-Stirling numbers, we derive
some interesting q-Stirling numbers identities associated with Carlitz q-Bernoulli numbers.
Finally, we will prove the following formula:
βn,q n
n mq km 1
1
− q
nm−k
q
k
i0 idi
s1,q k, m−1n−m
d0 ···dk n−k
where s1,q k, m is the q-Stirling number of the first kind.
m1
,
m 1q
1.9
Taekyun Kim
3
2. q-Stirling numbers and Carlitz q-Bernoulli numbers
For m ∈ Z , we note that
βm,q Zp
xm
q dμq x X
xm
q dμq x.
2.1
From this formula, we derive
qqβ 1k − βk,q
β0,q 1,
⎧
⎨1 if k 1,
⎩0 if k > 1,
2.2
with the usual convention of replacing βi by βi,q . By the simple calculation of p-adic q-integral
on Zp , we see that
βn,q
n
1
1 − qn i0
n
i
−1i
i1
,
i 1q
2.3
n
where i n!/i!n−i! nn−1 · · · n−i1/i!. Let Ft be the generating function of Carlitz
q-Bernoulli numbers. Then we have
Ft ∞
tn
βn,q
n!
n0
pρ −1
1 x xq t
lim ρ q e
ρ→∞ p
q x0
∞
∞
n
n
k1
1
k t
−1
n
n!
k k 1q
n0 1 − q
k0
et/1−q
2.4
∞
−1k
k 1 tk
.
k 1q k!
k0 1 − q
k
From 2.4 we note that
Ft et/1−q et/1−q
∞
−1k
k1 1
− qk−1
k
1 − qk1
k
∞
tk
1
t
−1k
et/1−q
k−1 1 − qk1 k!
k!
k1 1 − q
∞
∞
−t q2n enq t 1 − q qn enq t .
n0
2.5
n0
Therefore, we obtain the following.
n
n
Lemma 2.1. Let Ft ∞
n0 Zp xq dμq xt /n!. Then one has
∞
∞
Ft −t q2n enq t 1 − q qn enq t .
n0
n0
2.6
4
Advances in Difference Equations
The q-Bernoulli polynomials in the variable x in Cp with |x|p ≤ 1 are defined by
βn,q x x tnq dμq t x tnq dμq x.
Zp
Thus we have
Zp
x 2.7
X
tnq dμq x
n
n
k0
k
n
n
k0
k
kx
xn−k
q q
q β xq
Zp
tkq dμq t
2.8
kx
xn−k
q q βk,q
x
From 2.7 we derive
x tnq dμq x βn,q x Zp
n
.
n
1
1 − qn k0
n
k
−1k qkx
k1
.
k 1q
2.9
Let Ft, x be the generating function of q-Bernoulli polynomials. By 2.9 we see that
Ft, x ∞
βn,q x
n0
e
t/1−q
tn
n!
∞
k 1 tk
q
−1
.
k
k 1q k!
k0 1 − q
1
2.10
k
kx
From 2.10 we note that
∞
∞
Ft, x −t q2nx enxq t 1 − q qn enxq t .
n0
2.11
n0
By 2.7 and 2.11 we easily see that
mk−1
q
m−1
qi βk,qm
i0
xi
m
βk,q x,
m ∈ N, k ∈ Z .
If we take x 0 in 2.12, then we have
m
n−1
m
nq βn,q βk,qn nkq qjk1 jn−k
q .
k
j0
k0
2.12
2.13
∗
Let us define new q-Bernoulli polynomials, βn,q
x, as follows:
F ∗ t, x Ft, x − 1 − q
∞
qn enxq t
n0
∞
−t q2nx enxq t
n0
∞ β∗ x
n,q
n0
n!
tn .
2.14
Taekyun Kim
5
In the special case x 0, we can also derive the definition of q-Bernoulli numbers as follows:
∞
F ∗ t F ∗ t, 0 ∗
βn,q
n0
tn
.
n!
2.15
From these generating functions, we note that
−
∞
q
∞
∞
n−1
tm−1
m−1
2l lq t
2l
.
q e
m q lq
m!
m1
l0
l0
2ln nlq t
e
l0
Note that −
∞
l0 q
2ln nlq t
e
∞
∞
l0 q
2l lq t
∗
n
βm,q
1/tF ∗ t, n − F ∗ t. Thus, we have
e
−
∗
βm,q
m0
2.16
∞
n−1
tm tm
m−1
2l
m q lq
.
m! m0
m!
l0
2.17
By comparing the coefficients on both sides in 2.17, we see that
n−1
∗
∗
βm,q
n − βm,q
m q2l lm−1
.
q
2.18
l0
Therefore, we obtain the following.
Proposition 2.2. For m, n ∈ N, one has
q − 1
n−1
ql lm
q l0
n−1
ql lm−1
q
l0
1 ∗
∗
.
βm,q n − βm,q
m
2.19
Now we consider the q-analogue of Jordan factor as follows:
xk,q xq x − 1q · · · x − k 1q
1 − qx 1 − qx−1 · · · 1 − qx−k1
.
1 − qk
2.20
The q-binomial coefficient is defined by
n
k
q
nq !
kq !n − kq !
1 − qn−1 · · · 1 − qn−k1
,
1 − q 1 − q2 · · · 1 − qk
1 − qn
2.21
where nq ! nq n − 1q · · · 2q 1q . The q-binomial formulas are known as
n
a bq
i−1
i1
n
i1
1 − bq
n n
k0
i−1 −1
∞
k0
k
q
k
n−k k
2 a
b ,
q
2.22
nk−1
k
k
b .
q
6
Advances in Difference Equations
The q-Stirling numbers of the first kind s1,q n, k and the second kind s2,q n, k are defined as
n n
−
s1,q n, lxlq , n 0, 1, 2, . . . ,
xn,q q 2
2.23
l0
n k
q 2 s2,q n, kxk,q ,
xnq n 0, 1, 2, . . .
2.24
k0
see 2, 3, 6. The values s1,q n, 1, n 1, 2, 3, . . . , and s2,q n, 2, n 2, 3, . . . , may be deduced
from the following recurrence relation:
s1,q n, k s1,q n − 1, k − 1 − n − 1q s1,q n − 1, k
2.25
see 2, 3, 6, for k 1, 2, . . . , n, n 1, 2, . . . , with initial conditions s1,q 0, 0 1, s1,q n, k 0 if
k > n. For k 1, it follows that
s1,q n, 1 −n − 1q s1,q n − 1, 1,
n 2, 3, . . . ,
2.26
and since s1,q 1, 1 1, we have s1,q n, 1 −1n−1 n − 1q !, n 1, 2, 3, . . .. The recurrence
relation for k 2 reduces to s1,q n, 2 n − 1q s1,q n − 1, 2 −1n−2 n − 2q !, n 3, 4, . . .. By
simple calculation, we easily see that
s1,q n 1, 2 − nq s1,q n, 2
−1n1 s1,q n 1, 2 −1n s1,q n, 2
−
−1n1
nq !
n − 1q !
nq !
−1n1
1
,
nq
−1n1 n − 1q !
2.27
nq !
n 2, 3, 4, . . . .
Thus we have
n−1
−1n s1,q n, 2 1
.
n − 1q !
kq
k1
This is equivalent to s1,q n, 2 −1n n − 1q !
n
m1
−1m1 q 2
m1
n−1
k1 1/kq .
2.28
It is easy to see that
n
m
n
k1
1
k q
−1k1 q 2
.
kq
m 1 q k1 kq k1
n1
2.29
From this, we derive
n
k1
k1
−1k1 q 2
⎛ ⎛
⎞
⎞
n
k1
n
−
1
n−1
1 ⎝ n
1
⎠
⎝qn−k
⎠
−
−1k1 q 2
kq
kq
k−1 q
k q
k q
k1
n
k
qn k1 2 n
−1 q
nq k1
k
q
n
q
.
nq
2.30
Taekyun Kim
Note that
n
k1 −1
7
k1
k
q
n
2
n
k q
−
n
k1
−1k1 q 2
k1
k0 −1
n
k q
kq
k
k
q
2
n
k q
1 1. Thus, we have
n−1
k1
−1k1 q 2
n−1 k1
k
q
kq
qn
.
nq
2.31
Continuing this process, we see that
n
k1
−1k1 q 2
n
k q
kq
k1
n
qk
.
kq
k1
The p-adic q-gamma function is defined as Γp,q n −1n
we have Γp,q x 1 Ep,q xΓp,q x, where
Ep,q x ⎧
⎨−x
2.32
1≤j<n,j,p1 jq . For
if |x|p 1,
q
⎩−1
all x ∈ Zp ,
2.33
if |x|p < 1.
Thus, we easily see that
log Γp,q x 1 log Ep,q x log Γp,q x.
2.34
From the differentiation on both sides in 2.34, we derive
Γp,q x 1
Γp,q x 1
Γp,q x
Γp,q x
x
Ep,q
Ep,q x
.
2.35
Continuing this process, we have
Γp,q x
Γp,q x
x−1 j
q
j
q
j1
log q Γp,q 1
.
q − 1 Γp,q 1
2.36
The classical Euler constant is known as γ Γ 1/Γ1. In 15, Kim defined the p-adic q-Euler
constant as
γp,q −
Γp,q 1
Γp,q 1
.
2.37
Therefore, we obtain the following.
Theorem 2.3. For x ∈ Zp , one has
x−1
k1
k1
−1k1 q 2
x−1 k
kq
q
q − 1 Γp,q x
− γp,q .
log q Γp,q x
From 2.9, 2.21, 2.23, and 2.24, we derive the following theorem.
2.38
8
Advances in Difference Equations
Theorem 2.4. For n, k ∈ Z , one has
βn,q
n
1
n
1 − q l0
n
l
l
−1
l
q − 1
k
k
l k
k0
s1,q k, mβm,q ,
2.39
q m0
where s1,q k, m is the q-Stirling number of the first kind.
By simple calculation, we easily see that
n
qnt tq q − 1 1
n
n
−1m 1 − qm tm
q
m
m0
n
k
k 2 n
q − 1 q
tk,q
k q
k0
n
k
k n
q − 1
s1,q k, mtm
q
k
m0
k0
q
n
n
k n
q − 1
s1,q k, m tm
q .
k
m0 km
q
2.40
Thus we note
q dμq t nt
Zp
n
m0
n
q − 1
k
n
k
km
s1,q k, m βm,q .
2.41
q
From the definition of p-adic q-integral on Zp , we also derive
q dμq t nt
Zp
n
n
m0
m
q − 1m βm,q .
2.42
By comparing the coefficients on both sides of 2.41 and 2.42, we see that
n
m
m
q − 1 n
q − 1
km
k
n
k
s1,q k, m.
2.43
q
Therefore, we obtain the following.
Theorem 2.5. For n ∈ N, m ∈ Z , one has
n
n
−mk n
q − 1
s1,q k, m.
k q
m
km
2.44
From Theorem 2.5, we can also derive the following interesting formula for q-Bernoulli
numbers.
Taekyun Kim
9
Theorem 2.6. For n ∈ Z , one has
n
n
m1
1
−mk n
βn,q q − 1
s1,q k, m −1m
.
m 1q
1 − qn m0 km
k q
From the definition of q-binomial coefficient, we easily derive
x1
x
x
n
q
n q
n q
n−1 q
x
x
qx−n
.
n−1 q
n q
By 2.46, we see that
x
Zp
n
dμq x q
−1n n1− n1 2 .
q
n 1q
From the definition of q-Stirling number of the first kind, we also note that
x
xn,q dμq x nq !
dμq x
Zp
Zp n q
n
n − s1,q n, kβk,q .
q 2
2.45
2.46
2.47
2.48
k0
By using 2.47 and 2.48, we see
−1n
qnq !
n 1q
n
s1,q n, kβk,q .
2.49
k0
From 2.24 and 2.48, we derive
βn,q q
n
s2,q n, k−1k
k0
kq !
k 1q
.
2.50
,
2.51
Therefore, we obtain the following.
Theorem 2.7. For n ∈ Z , one has
βn,q q
n
s2,q n, k−1k
k0
kq !
k 1q
where s2,q n, k is the q-Stirling number of the second kind.
It is easy to see that
n
k
q
q
k
i0 idi
.
2.52
d0 ···dk n−k
By Theorem 2.4, we have the following.
Theorem 2.8. For n ∈ Z , one has
n n
k
1
m1
βn,q q i0 idi s1,q k, m−1n−m
,
nm−k
m
1q
m0 km 1 − q
d0 ···dk n−k
where s1,q k, m is the q-Stirling number of the first kind.
2.53
10
Advances in Difference Equations
Acknowledgment
This paper was supported by the research fund of Kangwoon University.
References
1 C. Adiga and N. Anitha, “On some continued fractions of Ramanujan,” Advanced Studies in Contemporary Mathematics, vol. 12, no. 1, pp. 155–162, 2006.
2 M. Cenkci, Y. Simsek, and V. Kurt, “Further remarks on multiple p-adic q-L-function of two variables,”
Advanced Studies in Contemporary Mathematics, vol. 14, no. 1, pp. 49–68, 2007.
3 M. Cenkci and M. Can, “Some results on q-analogue of the Lerch zeta function,” Advanced Studies in
Contemporary Mathematics, vol. 12, no. 2, pp. 213–223, 2006.
4 L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, no. 4, pp. 987–
1000, 1948.
5 L. Carlitz, “q-Bernoulli and Eulerian numbers,” Transactions of the American Mathematical Society,
vol. 76, no. 2, pp. 332–350, 1954.
6 A. S. Hegazi and M. Mansour, “A note on q-Bernoulli numbers and polynomials,” Journal of Nonlinear
Mathematical Physics, vol. 13, no. 1, pp. 9–18, 2006.
7 T. Kim and C. Adiga, “On the q-analogue of gamma functions and related inequalities,” Journal of
Inequalities in Pure and Applied Mathematics, vol. 6, no. 4, article 118, p. 4, 2005.
8 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299,
2002.
9 T. Kim, “On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Zp at
q −1,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 779–792, 2007.
10 T. Kim, “On a q-analogue of the p-adic log gamma functions and related integrals,” Journal of Number
Theory, vol. 76, no. 2, pp. 320–329, 1999.
11 T. Kim, “On p-adic q-L-functions and sums of powers,” Discrete Mathematics, vol. 252, no. 1–3, pp.
179–187, 2002.
12 T. Kim, S. D. Kim, and D.-W. Park, “On uniform differentiability and q-Mahler expansions,” Advanced
Studies in Contemporary Mathematics, vol. 4, no. 1, pp. 35–41, 2001.
13 T. Kim, “A note on the q-multiple zeta function,” Advanced Studies in Contemporary Mathematics, vol. 8,
no. 2, pp. 111–113, 2004.
14 T. Kim, “Sums of powers of consecutive q-integers,” Advanced Studies in Contemporary Mathematics,
vol. 9, no. 1, pp. 15–18, 2004.
15 T. Kim, “A note on p-adic invariant integral in the rings of p-adic integers,” Advanced Studies in Contemporary Mathematics, vol. 13, no. 1, pp. 95–99, 2006.
16 T. Kim, “A note on some formulae for the q-Euler numbers and polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 9, no. 2, pp. 227–232, 2006.
17 N. Koblitz, “q-extension of the p-adic gamma function,” Transactions of the American Mathematical Society, vol. 260, no. 2, pp. 449–457, 1980.
18 H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, “A note on p-adic q-Euler measure,” Advanced
Studies in Contemporary Mathematics, vol. 14, no. 2, pp. 233–239, 2007.
19 M. Schork, “Ward’s “calculus of sequences”, q-calculus and the limit q→ − 1,” Advanced Studies in
Contemporary Mathematics, vol. 13, no. 2, pp. 131–141, 2006.
20 Y. Simsek, “On p-adic twisted q − L-functions related to generalized twisted Bernoulli numbers,” Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340–348, 2006.
21 Y. Simsek, “Generalized Dedekind sums associated with the Abel sum and the Eisenstein and Lambert
series,” Advanced Studies in Contemporary Mathematics, vol. 9, no. 2, pp. 125–137, 2004.
22 Y. Simsek, “Twisted h, q-Bernoulli numbers and polynomials related to twisted h, q-zeta function
and L-function,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 790–804, 2006.
23 H. M. Srivastava, T. Kim, and Y. Simsek, “q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp.
241–268, 2005.