Volume 2008, Article ID 723615,9pages doi:10.1155/2008/723615
Research Article
On the Distribution of the
q
-Euler Polynomials and
the
q
-Genocchi Polynomials of Higher Order
Leechae Jang1 and Taekyun Kim2
1Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, South Korea
2Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
Correspondence should be addressed to Leechae Jang,[email protected]
Received 19 March 2008; Accepted 23 October 2008
Recommended by L´aszl ´o Losonczi
In 2007 and 2008, Kim constructed theq-extension of Euler and Genocchi polynomials of higher order and Choi-Anderson-Srivastava have studied theq-extension of Euler and Genocchi numbers of higher order, which is defined by Kim. The purpose of this paper is to give the distribution of extended higher-orderq-Euler andq-Genocchi polynomials.
Copyrightq2008 L. Jang and T. 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
The Euler numbersEnand polynomialsEnxare defined by the generating function in the complex number field as
2
et1 ∞
n0 Ent
n
n!
|t|< π,
2
et1ext ∞
n0 Enxt
n
n!
|t|< π,
1.1
cf. 1–4. The Bernoulli numbersBn and polynomialsBnxare defined by the generating function as
t et−1
∞
n0 Bnt
n
n!,
t et−1ext
∞
n0 Bnxt
n
n!,
cf.5–8. The Genocchi numbersGn and polynomialsGnxare defined by the generating function as
2t et1
∞
n0 Gnt
n
n!,
2t et1e
xt∞
n0 Gnxt
n
n!,
1.3
cf.9,10. It satisfiesG00, G11, . . . ,and forn≥1,
Gn2n
Bn
1 2
−Bn
. 1.4
Letp be a fixed odd prime number. Throughout this paper, Zp,Qp,and Cp will be, respectively, the ring of p-adic rational integers, the field of p-adic rational numbers and the p-adic completion of the algebraic closure of Qp. The p-adic absolute value in Cp is normalized so that|p|p 1/p. When one talks ofq-extension,qis variously considered as an indeterminate, a complex numberq∈Cor ap-adic numberq∈Cp.Ifq∈C,one normally assumes|q|<1.Ifq∈Cp,one normally assumes|1−q|p<1.We use the notation
xq 1−q
x
1−q, x−q
1−−qx
1q , 1.5
cf.1–5,9–23for allx∈Zp.For a fixed odd positive integerdwithp, d 1,set
XXdlim← n
Z
dpnZ, X1Zp,
X∗ 0<a<dp a,p1
adpZp
,
adpnZ p
x∈X|x≡amoddpn ,
1.6
wherea∈Zlies in 0≤a < dpn.For anyn∈N,
μq
adpnZp
qa dpn
q
1.7
is known to be a distribution onX,cf.1–5,9–23.
We say thatf is uniformly differentiable function at a pointa ∈ Zpand denote this property byf∈UDZp,if the difference quotients
Ffx, y fx−fy
x−y 1.8
have a limitlfaasx, y → a, a,cf.4.
Thep-adicq-integral of a functionf∈UDZpwas defined as
Iqf
Zp
fxdμqx lim n→ ∞
1 pn
q pn−1
x0
fxqx, 1.9
I−qf
Zp
fxdμ−qx nlim→ ∞ 1
pn q
pn−1
x0
cf.14. In1.10, whenq → 1,we derive
I−1
f1I−1f 2f0, 1.11
wheref1x fx1.If we takefx etx,then we havef1x etx1etxet.From1.11, we obtain
I−1
etx
Zp
etxdμ−1x
2
et1 ∞
n0 Ent
n
n!. 1.12
In view of1.10, we can consider theq-Euler numbers as follows:
I−q
etxq
Zp
etxqdμ−qx ∞
n0 En,qt
n
n!. 1.13
By1.12and1.13, we obtain the followings.
Lemma 1.1. Forn∈N,
En Gn1
n1. 1.14
Proof. We note that
tI−1
etx 2t
et1 ∞
n0 Gnt
n
n! ∞
n1 Gnt
n
n! ∞
n0 Gn1 n1
tn1 n! ,
tI−1
etx∞ n0
Zp
xndμ −1xt
n1 n! .
1.15
From1.15, we have
Gn1 n1
Zp
xndμ−1x En. 1.16
The purpose of this paper is to give the distribution of extended higher orderq-Euler and
q-Genocchi polynomials. In 24, Choi-Anderson-Srivastava have studied the q-extension of the Apostol-Euler polynomials of ordern,and the multiple Hurwitz zeta functionssee 24. Actually, their results and definitions are not newsee18,20and the definition of the Apostol-Bernoulli numbers in their paper are exactly the same as the definition of the
q-extension of Genocchi numbers. Finally, they conjecture that the followingq-distribution relation holds:
mq
k−1m−1
j0
−wjEk,qnm,wm
xj m
Ek,q,wn x 1.17
2. Weightedq-Genocchi number of higher order
In this section, we assume thatq∈Cpwith|1−q|p < 1 orq∈Cwith|q|<1.Fork ∈Nand
w∈Cpwith|1−w|p<1,we define the weightedq-Euler numbers of orderkas follows:
En,q,wk
Zp
· · ·
Zp
qkj1k−jxjwx1···xkx1· · ·x k
n qdμ−q
x1· · ·dμ−q
xk
. 2.1
We note thatq-binomial coefficient is defined by
n k
q
nqn−1q· · ·n−k1q kq
, 2.2
cf.20. From2.1, we obtain the following theorem.
Lemma 2.1. Fork∈N, n∈N∪ {0}andw∈Cpwith|1−w|p<1,one has
En,q,w 2k kq ∞
m0
mk−1
m
q
−1mwmqmmn
q. 2.3
Proof. From2.1, we have
En,q,wk
Zp
· · ·
Zp
qkj1k−jxjwx1···xkx1· · ·x k
n qdμ−q
x1· · ·dμ−q
xk
lim N→ ∞
1 pNk
−q pN−1
x1,...,xk0
qkj1k−jxjwx1···xkx1· · ·x k
n q−q
x1···xk
2 k q 2k
1
1−qn Nlim→ ∞ pN−1
x1,...,xk0
qkj1k−j1xj−1x1···xk
×wx1···xk n
l0
n l
−1lqlx1···xk
2 k q 2k
1 1−qn
n
l0
n l
−1l 2k Πk
j11qljw
2k q
1 1−qn
n
l0
n l
−1l∞ m0
mk−1
m
q
−1mqlmqmwm
2k q
∞
m0
mk−1
m
q
−1mqlmqmwm 1 1−qn
n
l0
n l
−1lqlm
2k q
∞
m0
mk−1
m
q
−1mqlmqmwmm q.
Now we consider the following generating functions:
Fq,wk t
∞
n0 En,q,wk t
n
n!
∞ n0
2k q
∞
m0
mk−1
m
q −1m
wmqmmnq
2k q
∞
m0
mk−1
m
q −1m
wmqmemqt.
2.5
By2.5, we can define the weightedq-Genocchi numbers of orderk:
Tq,wk t tkFq,wk t
∞
n0 Gn,q,wk t
n
n!. 2.6
From2.1,2.2, and2.6, we note that
G0k,q,w G1k,q,w · · ·Gkk−1,q,w0,
tk∞ n0
En,q,wk t n
n! ∞
nk
Gn,q,wk t n
n!.
2.7
Thus, we obtain
∞
n0 En,q,wk t
n
n! ∞
nk
Gn,q,wk t n−k
n!
∞ nk
Gnkk,q,w t
n
nk!
∞ nk
Gnkk,q,w 1 mk−1
m
tn
n!.
2.8
From 2.8, we obtain the following recurrsion relation between q-Euler and q-Genocchi numbers of orderk.
Theorem 2.2. Fork∈N, n∈N∪ {0}andw∈Cpwith|1−w|p<1,one has
mk k
k!En,q,wk Gnkk,q,w. 2.9
Fork∈N,we also define the weightedq-Euler polynomials of orderkas follows:
En,q,wk x
Zp
· · ·
Zp
qkj1k−jxjwx1···xkxx1· · ·x k
n qdμ−q
x1· · ·dμ−q
xk
From2.9, we obtain the following theorem.
Theorem 2.3. Fork∈N, n∈N∪ {0}andw∈Cpwith|1−w|p<1,one has
En,q,wk x 2kq ∞
m0
mk−1
m
q −1m
wmqmxmnq. 2.11
Proof.
En,q,wk x lim N→ ∞
1 pNk
−q pN−1
x1,...,xk0
qkj1k−jxjwx1···xkxx1· · ·x k
n q−q
x1···xk
2 k q 2k
1 1−qn
n
l0
n l
−1l
qlx lim N→ ∞
pN−1
x1,...,xk0
qkj1k−jl1xj−1x1···xkwx1···xk
2 k q 2k
1 1−qn
n
l0
n l
−1lqlx 2k Πk
j1
1qljw
2k q
1 1−qn
n
l0
n l
−1l
qlx
∞
m0
mk−1
m
q −1m
qlmqmwm
2k q
∞
m0
mk−1
m
q −1m
qlmqmwmxmq.
2.12
From2.11, we consider the following generating functions:
Fq,wk t, x
∞
n0
En,q,wk xt
n
n!
∞ n0
2k q
∞
m0
mk−1
m
q −1m
wmqmxmnq
2k q
∞
m0
mk−1
m
q −1m
wmqmexmqt.
2.13
By2.13, we can define the weightedq-Genocchi polynomials of orderkas follows:
Tq,wk t, x tkFq,wk t, x
∞
n0
Gn,q,wk xt
n
n!. 2.14
From2.14, we note that
G0k,q,w x G1k,q,w · · ·Gkk−1,q,wx 0,
tk∞ n0
En,q,wk xt
n
n! ∞
nk
Gn,q,wk xt
n
n!.
By comparing the coefficients on both sides, we see that
∞
n0
En,q,wk xt
n
n! ∞
nk
Gn,q,wk xt
n−k
n!
∞ nk
Gnkk,q,wx t
n
nk!
∞ nk
Gnkk,q,wx 1 mk−1
m
tn
n!.
2.16
From 2.16, we obtain the following recursion relation between weighted q-Euler and weightedq-Genocchi polynomials of orderk.
Theorem 2.4. Fork∈N, n∈N∪ {0}andw∈Cpwith|1−w|p<1,one has
mk k
k!En,q,wk x Gnkk,q,wx. 2.17
Corollary 2.5. Fork∈N, n∈N∪ {0}andw∈Cpwith|1−w|p<1,one has
Gnkk,q,wx k!
nk k
2k q 1−qn
n
l0
n l
−1lqxl 1 Πk
j1
1qljw
k!
nk k
2k q
∞
m0
mk−1
m
q −1m
wmqmxmnq.
2.18
Letd∈Nwithd≡1mod2.Then we note that
En,q,wk x
Zp
· · ·
Zp
qkj1k−jxjwx1···xkxx1· · ·xkn qdμ−q
x1· · ·dμ−q
xk
d m q dk−q
d−1
i1,...,ik0
qkkj1ij−kj2j−1ij−1kj1ijwi1···ik
×
Zp
· · ·
Zp
⎡
⎣x
k
j1ij
d
k
j1 xj
⎤ ⎦
m
qd
qd
k
j1k−jxj
wdx1···xk
×dμ−qd
x1· · ·dμ−qd
xk
d m q dk−q
d−1
i1,...,ik0
qkkj1ij−kj2j−1ij−1kj1ijEk m,qd,wd
xx1· · ·xk
d
.
2.19
Theorem 2.6 Distribution for higher order q-Euler polynomials. For d ∈ N with d ≡
1mod 2, n∈N∪ {0}andw∈Cpwith|1−w|p<1,one has
En,q,wk x d
m q dk−q
d−1
i1,...,ik0
qkkj1ij−kj2j−1ij−1kj1ijEk m,qd,wd
xx1· · ·xk
d
. 2.20
Fork∈N, w∈Cwith|w|<1,we easily see that
Fq,wk t, x 2kq ∞
m0
mk−1
m
q −1m
wmqmexmqt
∞
m0
Em,q,wk xt m
m!. 2.21
Thus we have
En,q,wk x d
n
dtnF k
q,wt, x 2kq ∞
m0
−1m
qmwmxmnq
mk−1
m
q
2.22
Definition 2.7. Fors∈C, k∈Nandw∈Cwith|w|<1,one has
ζq,w,Ek s, x 2kq ∞
m0
−1m
wmqm
mk−1
m
q mxsq
. 2.23
Note thatζq,w,Ek s, xis analytic function in the whole complexs-plane. From2.23, we derive the following.
Theorem 2.8. Forn∈N∪ {0}, k∈Nandw∈Cpwith|1−w|p<1,one has
ζq,w,Ek −n, x En,q,wk x. 2.24
Acknowledgments
The present research has been conducted by the research Grant of Kwangwoon University in 2008. The authors express their gratitude to referees for their valuable suggestions and comments.
References
1 L.-C. Jang, S.-D. Kim, D.-W. Park, and Y.-S. Ro, “A note on Euler number and polynomials,”Journal of Inequalities and Applications, vol. 2006, Article ID 34602, 5 pages, 2006.
2 T. Kim, “On Euler-Barnes multiple zeta functions,”Russian Journal of Mathematical Physics, vol. 10, no. 3, pp. 261–267, 2003.
3 T. Kim, “A note onp-adicq-integral onZp associated withq-Euler numbers,”Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 133–137, 2007.
4 T. Kim, “On p-adic interpolating function for q-Euler numbers and its derivatives,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 598–608, 2008.
5 L. Carlitz, “q-Bernoulli numbers and polynomials,”Duke Mathematical Journal, vol. 15, no. 4, pp. 987– 1000, 1948.
6 Y. Simsek, V. Kurt, and D. Kim, “New approach to the complete sum of products of the twisted
7 H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on sum of products of h, q-twisted Euler polynomials and numbers,” Journal of Inequalities and Applications, vol. 2008, Article ID 816129, 8 pages, 2008.
8 H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation functions of higher-orderq-Euler numbers and their applications,”Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008.
9 T. Kim, L.-C. Jang, and H. K. Pak, “A note onq-Euler and Genocchi numbers,”Proceedings of the Japan Academy. Series A, vol. 77, no. 8, pp. 139–141, 2001.
10 T. Kim, “On the multipleq-Genocchi and Euler numbers,”Russian Journal of Mathematical Physics, vol. 15, no. 4, pp. 481–486, 2008.
11 M. Cenkci and M. Can, “Some results onq-analogue of the Lerch zeta function,”Advanced Studies in Contemporary Mathematics, vol. 12, no. 2, pp. 213–223, 2006.
12 M. Cenkci, Y. Simsek, and V. Kurt, “Further remarks on multiplep-adicq-l-function of two variables,” Advanced Studies in Contemporary Mathematics, vol. 14, no. 1, pp. 49–68, 2007.
13 T. Kim, “q-Volkenborn integration,”Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.
14 T. Kim, “Analytic continuation of multipleq-zeta functions and their values at negative integers,” Russian Journal of Mathematical Physics, vol. 11, no. 1, pp. 71–76, 2004.
15 T. Kim, “Power series and asymptotic series associated with theq-analog of the two-variablep-adic
L-function,”Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 186–196, 2005.
16 T. Kim, “Multiplep-adicL-function,”Russian Journal of Mathematical Physics, vol. 13, no. 2, pp. 151– 157, 2006.
17 T. Kim, “On the analogs of Euler numbers and polynomials associated withp-adicq-integral onZpat
q−1,”Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 779–792, 2007.
18 T. Kim, “q-Euler numbers and polynomials associated withp-adicq-integrals,”Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007.
19 T. Kim, “On p-adic interpolating function for q-Euler numbers and its derivatives,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 598–608, 2008.
20 T. Kim, “q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,” Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008.
21 T. Kim, M.-S. Kim, L. Jang, and S.-H. Rim, “Newq-Euler numbers and polynomials associated with
p-adicq-integrals,”Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 243–252, 2007.
22 H. Ozden and Y. Simsek, “A new extension ofq-Euler numbers and polynomials related to their interpolation functions,”Applied Mathematics Letters, vol. 21, no. 9, pp. 934–939, 2008.
23 H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, “A note onp-adicq-Euler measure,”Advanced Studies in Contemporary Mathematics, vol. 14, no. 2, pp. 233–239, 2007.
