On the Symmetric Properties of Higher-Order Twisted q-Euler Numbers and Polynomials

(1)

doi:10.1155/2010/765259

Research Article

On the Symmetric Properties of Higher-Order Twisted q-Euler Numbers and Polynomials

Eun-Jung Moon,

¹

Seog-Hoon Rim,

²

Jeong-Hee Jin,

¹

and Sun-Jung Lee

¹

1Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea

2Department of Mathematics Education, Kyungpook National University, Daegu 702-701, South Korea

Correspondence should be addressed to Seog-Hoon Rim,[email protected] Received 14 December 2009; Accepted 19 March 2010

Academic Editor: Panayiotis Siafarikas

Copyrightq2010 Eun-Jung Moon 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.

In 2009, Kim et al. gave some identities of symmetry for the twisted Euler polynomials of higher- order, recently. In this paper, we extend our result to the higher-order twistedq-Euler numbers and polynomials. The purpose of this paper is to establish various identities concerning higher- order twistedq-Euler numbers and polynomials by the properties ofp-adic invariant integral on Zp. Especially, ifq1, we derive the result of Kim et al.2009.

1. Introduction

Letp be a fixed odd prime number. Throughout this paper, the symbolsZ, Zp,Qp,C,and Cp will denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of Qp, respectively. Let Nbe the set of natural numbers and Z N

{0}. Letvp be the normalized exponential valuation ofCpwith|p|pp^−v^p^p1/p.

When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ C, or a p-adic number q ∈ Cp. Ifq ∈ C,one normally assumes that |q| < 1.

Ifq∈Cp, then we assume that|q−1|p< p^−1/p−1so thatq^xexpxlogqfor eachx∈Zp.We use the following notation:

x_q 1−q^x

1−q , x_−q 1−

−q_x

1q ∀x∈Zp. 1.1

(2)

For a fixed positive integerdwithp, d 1, set

X Xd lim←n−Z

dpⁿZ , X1Zp, X^∗

0<a<dp, a,p1

adpZp

,

adpⁿZp

x∈X|x≡a

moddpⁿ ,

1.2

wherea∈Zsatisfies the condition 0≤a < dpⁿ.For anyn∈N,

μq

adpⁿZp

q^a

dpⁿ _q 1.3

see1–13is known to be a distribution onX.

We say thatfis a uniformly diﬀerentiable function ata∈Zpand denote this property byf ∈UDZpif the diﬀerence quotients

Ff

x, y

fx−f y

x−y 1.4

have a limitfaasx, y → a, a.

Forf∈UDZp,the fermionicp-adic invariantq-integral onZpis defined as

I−q f

Zp

fxdμ−qx lim

n→ ∞

1 pⁿ _−q

pⁿ−1 x0

fx

−q_x

1.5

see14. Let us define the fermionicp-adic invariant integral onZpas follows:

I−1 f

lim

q→1I−q f

Zp

fxdμ−1x lim

n→ ∞ pⁿ−1

x0

fx−1^x 1.6

see1–12,14–20. From the definition ofq-integral, we have I₋₁

f1

I₋₁ f

2f0, wheref1x fx1. 1.7

(3)

Forn∈N, letTpbe thep-adic locally constant space defined by

Tp

n≥1

Cpⁿ lim

n→ ∞Cpⁿ Cp^∞, 1.8

whereCpⁿ {ζ∈Cp|ζ^pⁿ 1 for somen≥0}is the cyclic group of orderpⁿ.

It is well known that the twistedq-Euler polynomials of orderkare defined as

e^xt 2

e^tζq1 _k

^∞

n0

E_n,ζ,q^k xtⁿ

n!, ζ∈Tp, 1.9

and E^k_n,ζ,q E^k_n,ζ,q0 are called the twisted q-Euler numbers of order k. When k 1, the polynomials and numbers are called the twisted q-Euler polynomials and numbers, respectively. When k 1 and q 1, the polynomials and numbers are called the twisted Euler polynomials and numbers, respectively. Whenk1,q1,andζ1, the polynomials and numbers are called the ordinary Euler polynomials and numbers, respectively.

In15, Kim et al. gave some identities of symmetry for the twisted Euler polynomials of higher order, recently. In this paper, we extend our result to the higher-order twisted q-Euler numbers and polynomials.

The purpose of this paper is to establish various identities concerning higher-order twistedq-Euler numbers and polynomials by the properties ofp-adic invariant integral on Zp. Especially, ifq1, we derive the result of15.

2. Some Identities of the Higher-Order Twisted q -Euler Numbers and Polynomials

Letw1, w2∈Nwithw1≡1mod 2andw2 ≡1mod 2.

Forζ∈Tpandm∈N, we set

R^mq w1, w2:ζ

Z^mp e^mⁱ¹^xⁱ^w²^xw¹^tζ^mⁱ¹^xⁱ^w¹q^mⁱ¹^xⁱ^w¹dμ−1x1· · ·dμ−1xm

Zpe^w¹^w²^xtζ^w¹^w²^xq^w¹^w²^xdμ₋₁x

×

Z^mp

e^mⁱ¹^xⁱ^w¹^yw²^tζ^mⁱ¹^xⁱ^w²q^mⁱ¹^xⁱ^w²dμ₋₁x1· · ·dμ₋₁xm,

2.1

where

Z^mp

fx1, . . . , xmdμ−1x1· · ·dμ−1xm

Zp

· · ·

Zp

m-times

fx1, . . . , xmdμ−1x1· · ·dμ−1xm. 2.2 In2.1, we note thatR^mq w1, w2:ζis symmetric inw1andw2.

(4)

From2.1, we derive that

R^mq w1, w2:ζ e^w¹^w²^xt

Z^mp

e^mⁱ¹^xⁱ^w¹^tζ^mⁱ¹^xⁱ^w¹q^mⁱ¹^xⁱ^w¹dμ−1x1· · ·dμ−1xm

×

Zpe^w²^x^m^tζ^w²^x^mq^w²^x^mdμ₋₁xm

Zpe^w¹^w²^xtζ^w¹^w²^xq^w¹^w²^xdμ₋₁x

×e^w¹^w²^yt

Z_pm−1

e^m−1ⁱ¹ ^xⁱ^w²^tζ^m−1ⁱ¹ ^xⁱ^w²q^m−1ⁱ¹ ^xⁱ^w²dμ−1x1· · ·dμ−1xm−1. 2.3 From the definition ofq-integral, we also see that

Zpm

e^mⁱ¹^xⁱ^w¹^tζ^mⁱ¹^xⁱ^w¹q^mⁱ¹^xⁱ^w¹dμ−1x1· · ·dμ−1xme^w¹^w²^xt

2 e^w¹^tζ^w¹q^w¹1

_m

e^w¹^w²^xt^∞

n0

E^m_n,ζw1,q^w¹w2xw1ntⁿ n! .

2.4

It is easy to see that

Zpe^xtζ^xq^xdμx

Zpe^w¹^xtζ^w¹^xq^w¹^xdμx ^w¹⁻¹

l0

−1^lζ^lq^le^lt^∞

k0

Tk,qw1−1 :ζt^k

k!, 2.5

whereTk,qw1−1 :ζ _w₁₋₁

l0 −1^lζ^lq^ll^k.

From2.3,2.4, and2.5, we can derive R^mq w1, w2:ζ

_∞

l0

E^m_l,ζw1,q^w¹w2xw^l₁t^l l!

_∞

k0

Tk,q^w²w1−1 :ζ^w² w^k₂t^k k!

_∞

i0

E_i,ζ^m−1w2,q^w²

w1yw2itⁱ i!

^∞

n0

⎧⎨

⎩ n

j0

n j

w₂^jw^n−j₁ E_n−j,ζ^m w1,q^w¹w2x j k0

Tk,q^w²w1−1 :ζ^w² j

k

E^m−1_j−k,ζw2,q^w²

w1y⎫

⎬

⎭tⁿ n!.

2.6

From the symmetry ofR^mq w1, w2:ζinw1andw2, we also see that R^mq w1, w2:ζ

^∞

n0

⎧⎨

⎩ n

j0

n j

w₁^jw^n−j₂ E_n−j,ζ^m w2,q^w²w1x j k0

Tk,q^w¹w2−1 :ζ^w¹ j

k

E^m−1_j−k,ζw1,q^w¹

w2y⎫

⎬

⎭tⁿ n!.

2.7

(5)

Comparing the coeﬃcients on the both sides of2.6and2.7, we obtain an identity for the twistedq-Euler polynomials of higher order as follows.

Theorem 2.1. Letw1, w2∈Nwithw1≡1mod 2andw2≡1mod 2.

Forn∈Zandm∈N,we have

n j0

n j

w^j₂w^n−j₁ E^m_n−j,ζw1,q^w¹w2x j k0

k

E_j−k,ζ^m−1w2,q^w²

w1y

ⁿ

j0

n j

w^j₁w₂^n−jE^m_n−j,ζw2,q^w²w1x j k0

k

E^m−1_j−k,ζw1,q^w¹

w2y .

2.8

Remark 2.2. Takingm1 andy0 inTheorem 2.1, we can derive the following identity:

n j0

n j

w^j₂w₁^n−jEn−j,ζ^w¹,q^w¹w2x j k0

k

ⁿ

j0

n j

w^j₁w₂^n−jEn−j,ζ^w²,q^w²w1x j k0

k

.

2.9

Moreover, if we takex 0 andy 0 in Theorem 2.1, then we have the following identity for the twistedq-Euler numbers of higher order.

Corollary 2.3. Letw1, w2 ∈Nwithw1≡1mod 2andw2 ≡1mod 2. Forn∈Zandm∈N, we have

n j0

n j

w^j₂w^n−j₁ E^m_n−j,ζw1,q^w¹

j k0

k

E^m−1_j−k,ζw2,q^w²

ⁿ

j0

n j

w^j₁w₂^n−jE^m_n−j,ζw2,q^w²

j k0

k

E_j−k,ζ^m−1w1,q^w¹.

2.10

We also note that takingm1 in Corollary 1 shows the following identity:

n j0

n j

w₂^jw^n−j₁ E_n−j,ζ^w1,q^w¹

j k0

k

ⁿ

j0

n j

w^j₁w^n−j₂ En−j,ζ^w²,q^w²

j k0

k

.

2.11

(6)

Now we will derive another interesting identities for the twistedq-Euler numbers and polynomials of higher order. From2.3, we can derive that

R^mq w1, w2:ζ

_w

1−1

i0

−1ⁱq^w²ⁱζ^w²ⁱ _∞

k0

E^m_k,ζw1,q^w¹

w2

w1iw2x

w₁^kt^k k!

_∞

l0

E_l,ζ^m−1w2,q^w²

w1y w₂^lt^l

l!

^∞

n0

_n

k0

n k

w₁^kw^n−k₂ E_n−k,ζ^m−1w2,q^w²

w1y^w¹⁻¹

i0

−1ⁱζ^w²ⁱq^w²ⁱE^m_k,ζw1,q^w¹

w2xw2

w1i tⁿ

n!. 2.12

From the symmetry ofR^mq w1, w2:ζinw1andw2, we see that

R^mq w1, w2:ζ ^∞

n0

_n

k0

n k

w^k₂w₁^n−kE^m−1_n−k,ζw1,q^w¹

w2y^w²⁻¹

i0

−1ⁱζ^w¹ⁱq^w¹ⁱE_k,ζ^mw2,q^w²

w1xw1

w2i tⁿ

n!. 2.13 Comparing the coeﬃcients on the both sides of2.12and 2.13, we obtain the following theorem which shows the relationship between the power sums and the twisted q-Euler polynomials.

Theorem 2.4. Letw1, w2 ∈Nwithw1 ≡1mod 2andw2 ≡1mod 2.Forn∈Z andm∈N, we have

n k0

n k

w^k₁w^n−k₂ E^m−1_n−k,ζw2,q^w²

w1y^w¹⁻¹

i0

w2x w2

w1i

ⁿ

k0

n k

w^k₂w₁^n−kE_n−k,ζ^m−1w1,q^w¹

w2y^w²⁻¹

i0

−1ⁱζ^w¹ⁱq^w¹ⁱE^m_k,ζw2,q^w²

w1xw1

w2i

.

2.14

Remark 2.5. Letm1 andy0 in Theorem 2.Then it follows that

n k0

n k

w^k₁w₂^n−k

w1−1 i0

−1ⁱζ^w²ⁱq^w²ⁱEk,ζ^w¹,q^w¹

w2xw2

w1i

ⁿ

k0

n k

w^k₂w₁^n−k

w2−1 i0

−1ⁱζ^w¹ⁱq^w¹ⁱEk,ζ^w²,q^w²

w1xw1

w2i

.

2.15

Moreover, if we takex0 andy0 inTheorem 2.4, then we have the following identity for the twistedq-Euler numbers of higher order.

(7)

Corollary 2.6. Letw1, w2 ∈Nwithw1 ≡1mod 2, w2 ≡ 1mod 2.Forn∈Z andm∈N,we have

n k0

n k

w₁^kw^n−k₂ E^m−1_n−k,ζw2,q^w² w1−1

i0

w2

w1i

ⁿ

k0

n k

w^k₂w₁^n−kE^m−1_n−k,ζw1,q^w¹ w2−1

i0

−1ⁱζ^w¹ⁱq^w¹ⁱE^m_k,ζw2,q^w²

w1

w2i

.

2.16

If we takem1 inCorollary 2.3, we derive the following identity for the twistedq-Euler polynomials:

forw1, w2∈Nwithw1≡1mod 2, w2≡1mod 2,andn∈Z,

n k0

n k

w₁^kw^n−k₂

w1−1 i0

−1ⁱζ^w²ⁱq^w²ⁱEk,ζ^w¹,q^w¹

w2

w1i

ⁿ

k0

n k

w₂^kw^n−k₁

w2−1 i0

−1ⁱζ^w¹ⁱq^w¹ⁱEk,ζ^w²,q^w²

w1

w2i

.

2.17

Remark 2.7. Ifq1,we can observe the result of15.

References

1 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.

2 T. Kim, J. Y. Choi, and J. Y. Sug, “Extended q-Euler numbers and polynomials associated with fermionicp-adicq-integral onZp,” Russian Journal of Mathematical Physics, vol. 14, no. 2, pp. 160–163, 2007.

3 T. Kim, “Symmetry of power sum polynomials and multivariate fermionicp-adic invariant integral onZp,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009.

4 T. Kim, “Onp -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 R. P. Agarwal and C. S. Ryoo, “Numerical computations of the roots of the generalized twistedq- Bernoulli polynomials,” Neural, Parallel & Scientific Computations, vol. 15, no. 2, pp. 193–206, 2007.

6 M. Cenkci, M. Can, and V. Kurt, “p-adic interpolation functions and Kummer-type congruences for q-twisted andq-generalized twisted Euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 9, no. 2, pp. 203–216, 2004.

7 F. T. Howard, “Applications of a recurrence for the Bernoulli numbers,” Journal of Number Theory, vol.

52, no. 1, pp. 157–172, 1995.

8 B. A. Kupershmidt, “Reflection symmetries of q-Bernoulli polynomials,” Journal of Nonlinear Mathematical Physics, vol. 12, supplement 1, pp. 412–422, 2005.

9 H. Ozden and Y. Simsek, “Interpolation function of theh, q-extension of twisted Euler numbers,”

Computers & Mathematics with Applications, vol. 56, no. 4, pp. 898–908, 2008.

10 L.-C. Jang, “A study on the distribution of twisted q-Genocchi polynomials,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 181–189, 2009.

11 M. Schork, “Ward’s “calculus of sequences”, q-calculus and the limitq → 1,” Advanced Studies in Contemporary Mathematics, vol. 13, no. 2, pp. 131–141, 2006.

12 H. J. H. Tuenter, “A symmetry of power sum polynomials and Bernoulli numbers,” The American Mathematical Monthly, vol. 108, no. 3, pp. 258–261, 2001.

13 T. Kim, “Note on the Eulerq-zeta functions,” Journal of Number Theory, vol. 129, no. 7, pp. 1798–1804, 2009.

(8)

14 T. Kim, “Symmetryp-adic invariant integral onZpfor Bernoulli and Euler polynomials,” Journal of Diﬀerence Equations and Applications, vol. 14, no. 12, pp. 1267–1277, 2008.

15 T. Kim, K. H. Park, and K.-W. Hwang, “On the identities of symmetry for theζ-Euler polynomials of higher order,” Advances in Diﬀerence Equations, vol. 2009, Article ID 273545, 9 pages, 2009.

16 T. Kim, “Note on the Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 131–136, 2008.

17 T. Kim, “Note on q-Genocchi numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 1, pp. 9–15, 2008.

18 T. Kim, “The modified q-Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 161–170, 2008.

19 T. Kim, “On aq-analogue of thep-adic log gamma functions and related integrals,” Journal of Number Theory, vol. 76, no. 2, pp. 320–329, 1999.

20 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.