q -Bernoulli Numbers Associated with q -Stirling Numbers

Advances in Difference Equations Volume 2008, Article ID 743295,10pages 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,[email protected] 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 sec- ond kinds. From the properties of q-Stirling numbers, we derive many interesting formu- las associated with Carlitz q-Bernoulli numbers. Finally, we will prove β_{n,q n}

m0_n

km1/

1−q^nm−k

d0···dkn−kq^ki0idis1,qk, m−1^n−mm1/m1_q,whereβ_n,qare called Carlitzq- Bernoulli numbers.

Copyrightq2008 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

Letpbe a fixed prime number. Throughout this paper, Zp,Qp,C, andCp will, respectively, denote the ring ofp-adic rational integers, the field ofp-adic rational numbers, the complex number field, and the completion of algebraic closure ofQp. Forda fixed positive integer with p, d 1, let

XXdlim

←−

N

Z

dp^NZ, X1Zp,

X^∗

0<a<dp a,p1

adpZp,

adp^NZp

x∈X|x≡a

moddp^N ,

1.1

where a ∈ Z lies in 0 ≤ a < dp^N, see 1–21. The p-adic absolute value in Cp is normal- ized so that|p|p 1/p. When one talks aboutq-extension,q is variously considered as an indeterminate, a complex number q ∈ C, or ap-adic number q ∈ Cp. If q ∈ Cp, then we

assume |q −1|_p < p^−1/p−1, so that q^x expxlogq for |x|p ≤ 1. We use the notation x_q x : q 1−q^x/1−q. Forf ∈ C¹Zp {f | f ∈ CZp}, let us start with the expressions

1 p^N

q

0≤j<p^N

q^jfj

0≤j<p^N

fjμq

jp^NZp

1.2

see6,8, representingq-analogue of Riemann sums forf. Thep-adicq-integral of a function f∈C¹Zpis defined by

X

fxdμqx

Zp

fxdμqx lim

N→∞

1 p^N

q p^N−1

x0

fxq^x 1.3

see8,22,23. Forf∈C¹Zp, it is easy to see that

Zp

fxdμqx p

≤pf1 1.4

see6–14, wheref1 sup{|f0|_p,sup_x/y|fx−fy/x−y|p}.Iffn→finC¹Zp, namely,fn−f1→0, then

Zp

fnxdμqx−→

Zp

fxdμqx 1.5

see6–10. Theq-analogue of binomial coefficient was known as ^x_n

q x_qx−1_q· · ·x

−n1_q/n_q!,wheren_q!_n

i1i_qsee1,5,6,10,11. From this definition, we derive x1

n

q

x

n−1

q

qⁿ x

n

q

q^x−n x

n−1

q

x

n

q

1.6

cf.6,10. Thus, we have

Zp

x n

qdμqx −1ⁿ/n1_qqⁿ¹⁻

_n1

2

.Iffx

k≥0ak,q x k

q

is theq-analogue of Mahler series of strictly differentiable functionf, then we see that

Zp

fxdμqx

k≥0

ak,q −1^k k1_qq^k1−

_k1

2

. 1.7

Carlitzq-Bernoulli numbersβk,qβkqcan be determined inductively by β0,q1, qqβ1^k−βk,q

⎧⎨

⎩

1 ifk1,

0 ifk >1, 1.8

with the usual convention of replacingβⁱ by βi,q see 2–4. In this paper, we study theq- Stirling numbers of the first and the second kinds. From theseq-Stirling numbers, we derive some interesting q-Stirling numbers identities associated with Carlitz q-Bernoulli numbers.

Finally, we will prove the following formula:

βn,qⁿ

mq

n km

1 1−q^nm−k

d0···dkn−k

q^ki0idis1,qk, m−1^n−m m1

m1_q, 1.9 wheres1,qk, mis theq-Stirling number of the first kind.

2.q-Stirling numbers and Carlitzq-Bernoulli numbers

Form∈Z, we note that

βm,q

Zp

x^m_qdμqx

X

x^m_qdμqx. 2.1

From this formula, we derive

β0,q1, qqβ1^k−βk,q

⎧⎨

⎩

1 ifk1,

0 ifk >1, 2.2

with the usual convention of replacingβⁱbyβi,q. By the simple calculation ofp-adicq-integral onZp, we see that

βn,q 1 1−qⁿ

n i0

n i

−1ⁱ i1

i1_q, 2.3 wheren

i

n!/i!n−i!nn−1· · ·n−i1/i!. LetFtbe the generating function of Carlitz q-Bernoulli numbers. Then we have

Ft ^∞

n0

βn,qtⁿ n!

lim

ρ→∞

1 p^ρ

q p^ρ−1

x0

q^xe^{xqt ∞}

n0

1 1−qⁿ

_∞

k0

n k

k1 k1_q−1^k

tⁿ n!

e^t/1−q ∞ k0

−1^k 1−q^k

k1 k1_q

t^k k!.

2.4

From2.4we note that Ft e^t/1−qe^t/1−q

∞ k1

−1^k 1−q^k−1

k 1−q^k1

t^k

k!e^t/1−q ∞ k1

−1^k 1−q^k−1

1 1−q^k1

t^k k!

−t^∞

n0

q²ⁿe^nqt 1−q^∞

n0

qⁿe^nqt.

2.5

Therefore, we obtain the following.

Lemma 2.1. LetFt _∞

n0

Zpxⁿ_qdμqxtⁿ/n!. Then one has Ft −t^∞

n0q²ⁿe^nqt 1−q^∞

n0qⁿe^nqt. 2.6

Theq-Bernoulli polynomials in the variablexinCpwith|x|p≤1 are defined by βn,qx

Zp

xtⁿ_qdμqt

X

xtⁿ_qdμqx. 2.7 Thus we have

Zp

xtⁿ_qdμqx ⁿ

k0

n k

x^n−k_q q^kx

Zp

t^k_qdμqt

ⁿ

k0

n k

x^n−k_q q^kxβk,q

q^xβ x_qn

.

2.8

From2.7we derive

Zp

xtⁿ_qdμqx βn,qx 1 1−qⁿ

n k0

n k

−1^kq^kx k1

k1_q. 2.9 LetFt, xbe the generating function ofq-Bernoulli polynomials. By2.9we see that

Ft, x ^∞

n0

βn,qxtⁿ n!

e^t/1−q ∞ k0

1

1−q^kq^kx−1^k k1 k1_q

t^k k!.

2.10

From2.10we note that

Ft, x −t^∞

n0

q^2nxe^nxqt 1−q^∞

n0

qⁿe^nxqt. 2.11

By2.7and2.11we easily see that m^k−1_q ^m−1

i0

qⁱβk,q^m

xi m

βk,qx, m∈N, k∈Z. 2.12 If we takex0 in2.12, then we have

n_qβn,q^m

k0

m k

βk,qⁿn^k_qⁿ⁻¹

j0

q^jk1j^n−k_q . 2.13

Let us define newq-Bernoulli polynomials,β^∗_n,qx, as follows:

F^∗t, x Ft, x−1−q^∞

n0

qⁿe^nxqt −t^∞

n0

q^2nxe^{nxqt ∞}

n0

βn,q^∗ x n! tⁿ.

2.14

In the special casex0, we can also derive the definition ofq-Bernoulli numbers as follows:

F^∗t F^∗t,0 ^∞

n0β^∗_n,qtⁿ

n!. 2.15

From these generating functions, we note that

−^∞

l0

q^2lne^nlqt∞

l0

q^2le^lqt∞

m1

m

n−1 l0

q^2ll^m−1_q t^m−1

m! . 2.16

Note that−_∞

l0q^2lne^nlqt_∞

l0q^2le^lqt 1/tF^∗t, n−F^∗t.Thus, we have ∞

m0

β_m,q^∗ n−β^∗_m,qt^m m!^∞

m0

m

n−1

l0

q^2ll^m−1_q t^m

m!. 2.17

By comparing the coefficients on both sides in2.17, we see that

β^∗_m,qn−β^∗_m,qm n−1

l0

q^2ll^m−1_q . 2.18

Therefore, we obtain the following.

Proposition 2.2. Form, n∈N, one has q−1ⁿ⁻¹

l0

q^ll^m_q ⁿ⁻¹

l0

q^ll^m−1_q 1 m

β^∗_m,qn−β_m,q^∗

. 2.19

Now we consider theq-analogue of Jordan factor as follows:

x_k,q x_qx−1_q· · ·x−k1_q

1−q^x

1−q^x−1

· · ·

1−q^x−k1

1−q^k . 2.20 Theq-binomial coefficient is defined by

n k

q

n_q! k_q!n−k_q!

1−qⁿ

1−qⁿ⁻¹

· · ·

1−q^n−k1 1−q

1−q²

· · ·

1−q^k , 2.21

wheren_q! n_qn−1_q· · ·2_q1_q.Theq-binomial formulas are known as n

i1

abq^{i−1 n}

k0

n k

q

q^k2a^n−kb^k,

n i1

1−bqⁱ⁻¹−1

^∞

k0

nk−1 k

q

b^k.

2.22

Theq-Stirling numbers of the first kinds1,qn, kand the second kinds2,qn, kare defined as x_n,qq⁻

_n

2

_n

l0

s1,qn, lx^l_q, n0,1,2, . . . , 2.23 xⁿ_q ⁿ

k0

q _k

2

s2,qn, kx_k,q, n0,1,2, . . . 2.24 see2,3,6. The valuess1,qn,1, n 1,2,3, . . . ,ands2,qn,2, n2,3, . . . ,may be deduced from the following recurrence relation:

s1,qn, k s1,qn−1, k−1−n−1_qs1,qn−1, k 2.25 see2,3,6, fork1,2, . . . , n,n1,2, . . . ,with initial conditionss1,q0,0 1,s1,qn, k 0 if k > n. Fork1, it follows that

s1,qn,1 −n−1_qs1,qn−1,1, n2,3, . . . , 2.26 and since s1,q1,1 1, we haves1,qn,1 −1ⁿ⁻¹n−1_q!, n 1,2,3, . . .. The recurrence relation fork2 reduces tos1,qn,2 n−1_qs1,qn−1,2 −1ⁿ⁻²n−2_q!, n 3,4, . . .. By simple calculation, we easily see that

−1ⁿ¹s1,qn1,2

n_q! −−1ⁿs1,qn,2

n−1_q! −1ⁿ¹s1,qn1,2−n_qs1,qn,2 n_q!

−1ⁿ¹−1ⁿ¹n−1_q! n_q! 1

n_q, n2,3,4, . . . .

2.27

Thus we have

−1ⁿs1,qn,2 n−1_q! ⁿ⁻¹

k1

1

k_q. 2.28 This is equivalent tos1,qn,2 −1ⁿn−1_q!_n−1

k11/k_q.It is easy to see that n

m1

−1^m1q

m1

2

n1 m1

q

m k1

1 k_q ⁿ

k1

−1^k1q

k1 2

n k

q

k_q. 2.29 From this, we derive

n k1

−1^k1q

k1

2 1

k_q

⎛

⎝ n

k

q

− n−1

k

q

⎞

⎠ⁿ

k1

−1^k1q

k1

2 1

k_q

⎛

⎝q^n−k n−1

k−1

q

⎞

⎠ qⁿ

n_q n k1

−1^k1q

k 2

n k

q

qⁿ n_q.

2.30

Note that_n

k1−1^k1q _k

2

n k

q−_n

k0−1^kq _k

2

n k

q11.Thus, we have n

k1

−1^k1q

k1 2

n k

q

k_q ⁿ⁻¹

k1

−1^k1q

k1 2

n−1 k

q

k_q qⁿ

n_q. 2.31 Continuing this process, we see that

n k1

−1^k1q

k1 2

n k

q

k_q ⁿ

k1

q^k

k_q. 2.32 Thep-adicq-gamma function is defined asΓp,qn −1ⁿ

1≤j<n,j,p1j_q.For allx∈Zp, we haveΓp,qx1 Ep,qxΓp,qx, where

Ep,qx

⎧⎨

⎩

−x_q if|x|p1,

−1 if|x|p<1. 2.33

Thus, we easily see that

logΓp,qx1 logEp,qx logΓp,qx. 2.34 From the differentiation on both sides in2.34, we derive

Γ_p,qx1

Γp,qx1 Γ_p,qx

Γp,qxE_p,q x

Ep,qx. 2.35

Continuing this process, we have Γ_p,qx Γp,qx

_x−1

j1

q^j j_q

logq

q−1Γ_p,q1

Γp,q1. 2.36 The classical Euler constant is known asγ Γ1/Γ1. In15, Kim defined thep-adicq-Euler constant as

γp,q−Γ_p,q1

Γp,q1. 2.37 Therefore, we obtain the following.

Theorem 2.3. Forx∈Zp, one has

x−1 k1

−1^k1q

k1 2

x−1 k

q

k_q q−1 logq

Γ_p,qx Γp,qx−γp,q

. 2.38

From2.9,2.21,2.23, and2.24, we derive the following theorem.

Theorem 2.4. Forn, k∈Z, one has βn,q 1

1−qⁿ n l0

n l

−1^ll

k0

q−1^k l

k

q

k m0

s1,qk, mβm,q, 2.39

wheres1,qk, mis theq-Stirling number of the first kind.

By simple calculation, we easily see that q^nt

t_qq−1 1n

ⁿ

m0

n m

−1^m1−q^mt^m_q

ⁿ

k0

q−1^kq

k 2

n k

q

t_k,q

ⁿ

k0

q−1^k n

k

q

k m0

s1,qk, mt^m_q

ⁿ

m0

_n

km

q−1^k n

k

q

s1,qk, m

t^m_q.

2.40

Thus we note

Zp

q^ntdμqt ⁿ

m0

_n

km

q−1^k n

k

q

s1,qk, m

βm,q. 2.41

From the definition ofp-adicq-integral onZp, we also derive

Zp

q^ntdμqt ⁿ

m0

n m

q−1^mβm,q. 2.42 By comparing the coefficients on both sides of2.41and2.42, we see that

n m

q−1^{m n}

km

q−1^k n

k

q

s1,qk, m. 2.43

Therefore, we obtain the following.

Theorem 2.5. Forn∈N, m∈Z, one has n

m

ⁿ

km

q−1^−mk n

k

q

s1,qk, m. 2.44

FromTheorem 2.5, we can also derive the following interesting formula forq-Bernoulli numbers.

Theorem 2.6. Forn∈Z, one has βn,q 1

1−qⁿ n m0

_n

km

q−1^−mk n

k

q

s1,qk, m

−1^m m1

m1_q. 2.45 From the definition ofq-binomial coefficient, we easily derive

x1 n

q

x

n−1

q

qⁿ x

n

q

q^x−n x

n−1

q

x

n

q

.

2.46

By2.46, we see that

Zp

x n

q

dμqx −1ⁿ n1_qqⁿ¹⁻

n1 2

. 2.47

From the definition ofq-Stirling number of the first kind, we also note that

Zp

x_n,qdμqx n_q!

Zp

x n

q

dμqx

q⁻

n 2n

k0

s1,qn, kβk,q.

2.48

By using2.47and2.48, we see

−1ⁿ qn_q! n1_q ⁿ

k0

s1,qn, kβk,q. 2.49 From2.24and2.48, we derive

βn,qq n k0

s2,qn, k−1^k k_q!

k1_q. 2.50 Therefore, we obtain the following.

Theorem 2.7. Forn∈Z, one has βn,qq

n k0

s2,qn, k−1^k k_q!

k1_q, 2.51 wheres2,qn, kis theq-Stirling number of the second kind.

It is easy to see that

n k

q

d0···dkn−k

q^ki0idi. 2.52

ByTheorem 2.4, we have the following.

Theorem 2.8. Forn∈Z, one has βn,qⁿ

m0

n km

1 1−q^nm−k

d0···dkn−k

q^ki0idis1,qk, m−1^n−m m1

m1_q, 2.53 wheres1,qk, mis theq-Stirling number of the first kind.

Acknowledgment

This paper was supported by the research fund of Kangwoon University.

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