IJMMS 2004:49, 2649–2651 PII. S0161171204402105 http://ijmms.hindawi.com
© Hindawi Publishing Corp.
AN EXTENSION OF q-ZETA FUNCTION
T. KIM, L. C. JANG, and S. H. RIM Received 27 February 2004
We will define the extension ofq-Hurwitz zeta function due to Kim and Rim (2000) and study its properties. Finally, we lead to a useful new integral representation for theq-zeta function.
2000 Mathematics Subject Classification: 11B68, 11S40.
1. Introduction. Let 0< q <1 and for any positive integerk, define itsq-analogue [k]q=(1−qk)/(1−q). LetCbe the field of complex numbers. Theq-zeta function due to T. Kim was defined as
ζq(h)(s)= ∞ n=1
qnh
[n]sq+(q−1)1−s+h 1−s
∞ n=1
qnh [n]s−q 1
(1.1)
for anys,h∈C(cf. [3,4]). This function can be considered on the spectral zeta function of the quantum group SUq(2)(cf. [2,4]). Also, theq-zeta functionζq(h)(s)was studied at negative integers (see [4]). In this note, we lead to a useful new integral representa- tion for theq-zeta functionζq(h)(s). Finally, we define the extension ofq-Hurwitz zeta function, and study its properties.
2. q-zeta functions. Forq∈Cwith |q|<1, we defineq-Bernoulli polynomials as follows:
Fq(h)(t,x)= ∞ n=0
β(h)n,q(x) n! tn
=e(1/(1−q))t ∞ j=0
j+h [j+h]q
(−1)jqjx 1
1−q jtj
j!
= −t ∞ l=0
ql(h+1)+xe[l+x]qt+(1−q)h ∞ l=0
qlhe[l+x]qt
(2.1)
forh∈Z,x∈C(cf. [2,4]). In the casex=0,β(h)n,q(=β(h)n,q(0))will be called theq-Bernoulli numbers (cf. [4]). By (2.1), we easily see that
2650 T. KIM ET AL.
β(h)n,q(x)= m j=0
m j
[x]n−jq qjxβ(h)j,q
= 1
1−q n n
j=0
n j
(−1)j j+h
[j+h]qqjx cf. [2]
,
(2.2)
where
n
j is a binomial coefficient.
Thus we note that
qh
qβ(h)+1n
−β(h)n,q=δ1,n, (2.3)
where we use the usual convention about replacing (β(h))n by β(h)n,q and δ1,n is the Kronecker symbol.
Example2.1.
β(2)0 = 2
[2], β(2)1 = −2q+1
[2][3], β(2)2 = 2q2
[3][4], β(2)3 = −q2(q−1)
2[3]q+q [3][4][5] , ....
(2.4) LetFq(h)(t)=∞
n=0(β(h)n,q/n!)tn. Then we easily see that Fq(h)(x,t)=e[x]qtFq(h)
qxt
= −t ∞ l=0
ql(h+1)+xe[l+x]qt+(1−q)h ∞ l=0
qlhe[l+x]qt. (2.5)
By (2.1) and (2.5), we note that
e−tFq(h)(−qt)=qt ∞ l=0
ql(h+1)e−[l+1]qt+(1−q)h ∞ l=0
qlhe−[l+1]qt. (2.6)
Thus we have 1 Γ(s)
∞
0 qhts−2e−tFq(h)(−qt)dt= ∞ n=1
qnh
[n]sq+(q−1)h+1−s 1−s
∞ n=1
qnh
[n]s−q 1. (2.7) Forh,s∈C, we define theq-zeta function as follows:
ζq(h)(s)= ∞ n=1
qnh
[n]sq+(q−1)1−s+h 1−s
∞ n=1
qnh [n]s−1
cf. [1,4]
. (2.8)
Note thatζq(h)(s)is a meromorphic function for Re(s) >1.
LetΓ(s)be the gamma function and letZbe the set of integers. By (2.3), (2.7), and (2.8), we obtain the following.
Forh,n(>1)∈Z, we have
ζq(h)(1−n)= −qh
qβ(h)+1n
n = −β(h)n,q
n . (2.9)
AN ANALOGUE OFq-ZETA FUNCTION 2651 Letxbe any nonzero positive real number. Then we define theq-analogue of Hurwitz zeta function as follows:
ζq(h)(s,x)= ∞ n=0
qnh
[n+x]sq+h+1−s 1−s (q−1)
∞ n=0
qnh [n+x]s−1q
(2.10)
fors,h∈C. By (2.5) and (2.10), we easily see that
ζ(h)q (s,x)= 1 Γ(s)
∞
0 ts−2Fq(h)(x,−t)dt. (2.11) Thus we obtain the following: forn∈N,h∈Z, we have
ζq(h)(1−n)= −β(h)n,q(x)
n (2.12)
because
ζ(h)q (s,x)= ∞ n=0
(−1)n
n! β(h)n,q(x) 1 Γ(s)
∞
0 ts+n−2dt. (2.13) Acknowledgments. This research was supported by Kyungpook National Univer- sity Research Team Fund, 2003. This paper was dedicated to Chung-Seo Park.
References
[1] T. Kim,On Euler-Barnes multiple zeta functions, Russian J. Math. Phys. 10(2003), no. 3, 261–267.
[2] ,Analytic continuation of multipleq-zeta functions and their values at negative inte- gers, Russ. J. Math. Phys.11(2004), no. 1, 71–76.
[3] ,q-Riemann zeta function, Int. J. Math. Math. Sci.2004(2004), no. 9–12, 599–605.
[4] T. Kim and S. H. Rim,Generalized Carlitz’sq-Bernoulli numbers in thep-adic number field, Adv. Stud. Contemp. Math. (Pusan)2(2000), 9–19.
T. Kim: Institute of Science Education, Kongju National University, Kongju 314-701, Korea E-mail address:[email protected]
L. C. Jang: Department of Mathematics and Computer Science, Konkuk University, Choongju 380-701, Korea
E-mail address:[email protected]
S. H. Rim: Department of Mathematics Education, Kyungpook National University, Daegu 702- 701, Korea
E-mail address:[email protected]
