Note on q-Extensions of Euler Numbers and Polynomials of Higher Order

Journal of Inequalities and Applications Volume 2008, Article ID 371295,9pages doi:10.1155/2008/371295

Research Article

Note on q-Extensions of Euler Numbers and Polynomials of Higher Order

Taekyun Kim,¹Lee-Chae Jang,²and Cheon-Seoung Ryoo³

1The School of Electrical Engineering and Computer Science (EECS), Kyungpook National University, Taegu 702-701, South Korea

2Department of Mathematics and Computer Science, KonKuk University, Chungju 143-701, South Korea

3Department of Mathematics, Hannam University, Daejeon 306-791, South Korea

Correspondence should be addressed to Cheon-Seoung Ryoo,[email protected] Received 1 November 2007; Accepted 22 December 2007

Recommended by Paolo Emilio Ricci

In 2007, Ozden et al. constructed generating functions of higher-order twistedh, q-extension of Euler polynomials and numbers, by usingp-adic,q-deformed fermionic integral onZp. By apply- ing their generating functions, they derived the complete sums of products of the twistedh, q- extension of Euler polynomials and numbers. In this paper, we consider the newq-extension of Eu- ler numbers and polynomials to be different which is treated by Ozden et al. From ourq-Euler num- bers and polynomials, we derive some interesting identities and we constructq-Euler zeta functions which interpolate the newq-Euler numbers and polynomials at a negative integer. Furthermore, we study Barnes-typeq-Euler zeta functions. Finally, we will derive the new formula for “sums of prod- ucts ofq-Euler numbers and polynomials” by using fermionicp-adic,q-integral onZp.

Copyrightq2008 Taekyun Kim 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.

1. Introduction and notations

Throughout this paper we use the following notations. ByZpwe denote the ring ofp-adic ra- tional integers,Qdenotes the field of rational numbers,Qpdenotes the field ofp-adic rational numbers,Cdenotes the complex number field, andCpdenotes the completion of algebraic clo- sure ofQp. Letνpbe the normalized exponential valuation ofCpwith|p|pp^−νppp⁻¹.When one talks ofq-extension,q is considered in many ways such as an indeterminate, a complex numberq∈C,orp-adic numberq∈Cp.Ifq∈C, one normally assumes that|q|<1.Ifq∈Cp, we normally assume that|q−1|_p< p^−1/p−1so thatq^xexpxlogqfor|x|p≤1.In this paper, we use the following notation:

x_q x:q 1−q^x

1−q 1.1

cf.1–5,22.

Hence, limq→1x xfor anyxwith|x|p≤1 in the presentp-adic case. Letdbe a fixed integer and letpbe a fixed prime number. For any positive integerN, we set

Xlim_←

N

Z dp^NZ

,

X^∗

0<a<dp

a,p1

adpZp

,

adp^NZp

x∈X|x≡a

moddp^N ,

1.2

wherea∈Zlies in 0≤a < dp^N. For any positive integerN, μ_q

adp^NZp q^a

dp^N

q

1.3

is known to be a distribution onXcf.1–20. From this distribution, we derive thep-adic, q-integral onZpas follows:

Zp

fxdμ_qx lim

N→∞

1 p^N

q p^N−1

x0

q^xfx, f∈UD Zp

, 1.4

see1–23.

Higher-order twisted Bernoulli and Euler numbers and polynomials are studied by many authors see for detail 1–21. In14Ozden et al. constructed generating functions of higher-order twistedh, q-extension of Euler polynomials and numbers, by usingp-adic, q-deformed fermionic integral onZp. By applying their generating functions, they derived the complete sums of products of the twistedh, q-extension of Euler polynomials and numbers, see14,15. In this paper, we consider the newq-extension of Euler numbers and polynomials to be different which is treated by Ozden et al. From ourq-Euler numbers and polynomials, we derive some interesting identities and we constructq-Euler zeta functions which interpo- late the newq-Euler numbers and polynomials at a negative integer. Furthermore, we study Barnes-typeq-Euler zeta functions. Finally, we will derive the new formula for “sums of prod- ucts ofq-Euler numbers and polynomials” by using fermionicp-adic,q-integral onZp. 2.q-extension of Euler numbers

In this section we assume thatq ∈Cwith|q| <1. Now we consider theq-extension of Euler polynomials as follows:

Fqx, t 2_q

qe^t1e^xt∞

n0

En,qx

n! tⁿ, |tlogq|< π. 2.1

Note that

limq→1F_qx, t Fx, t 2

e^t1e^xt∞

n0

E_nx

n! tⁿ. 2.2

In the special casex0,theq-Euler polynomialEn,q0 En,qwill be calledq-Euler numbers.

It is easy to see thatFqx, tis analytic function inC. Hence we have ∞

n0

E_n,qx

n! tⁿ 2_q

qe^t1e^xt 2_q^∞

n0

−1ⁿqⁿe^nxt. 2.3

If we take thekth derivative att0 on both sides in2.3, then we have Ek,qx 2_q^∞

n0

−1ⁿqⁿnx^k. 2.4

From2.4we can defineq-zeta function which interpolatingq-Euler numbers at negative in- teger as follows.

Fors∈C,we define

ζ_qs, x 2_q^∞

n0

−1ⁿqⁿ

nx^s, s∈C. 2.5 Note thatζ_qs, xis analytic in complexs-plane. If we take s −kk ∈ Z,then we have ζ_q−k, x E_k,qx.

By2.4and2.5, we obtain the following.

Theorem 2.1. Fork∈Z,

E_k,qx 2_q^∞

n0

−1ⁿqⁿnx^k. 2.6

LetF_q0, t F_qt.Then

2_qⁿ⁻¹

k0

−1^kq^ke^kt 2_q

1qe^t −2_q−1ⁿqⁿe^nt 1qe^t F_qt−−1ⁿqⁿF_qn, t.

2.7

From2.7, derive ∞

k0

2_qⁿ⁻¹

l0

−1^lq^ll^k t^k

k! ^∞

k0

Ek,q−−1ⁿqⁿEk,qnt^k

k!. 2.8

By comparing the coefficients on both sides in2.8, we obtain the following.

Theorem 2.2. Letn∈N, k∈Z. Ifn≡0 mod 2, then

Ek,q−qⁿEk,qn 2_qⁿ⁻¹

l0

−1^lq^ll^k. 2.9 Ifn≡1 mod 2, then

E_k,qqⁿE_k,qn 2_qⁿ⁻¹

l0

−1^lq^ll^k. 2.10 Forw₁, w₂, . . . , w_r ∈C,consider the multipleq-Euler polynomials of Barnes-type as follows:

F_q^r

w1, w2, . . . , wr |x, t

2^r_qe^xt qe^w1t1

qe^w2t1

· · ·

qe^wrt1 ^∞

n0

E_n,q

w₁, . . . , w_r|xtⁿ

n!, where max

1≤i≤r |witlogq|< π.

2.11

For x 0, E_n,qw1, . . . , w_r | 0 E_n,qw1, . . . , w_rwill be called the multiple q-Euler numbers of Barnes type. It is easy to see thatF_q^rw1, w2, . . . , wr |x, tis analytic function in the given region. From2.11, we derive

2^r_q ^∞

n1,...,nr0

−q^ri1nie^ri1niwixt∞

n0

En,q

w1, . . . , wr |xtⁿ

n!. 2.12

By thekth differentiation on both sides in2.12, we see that 2^r_q ^∞

n1,...,nr0

−q^ri1ni _r

i1

n_iw_ix _k

E_k,q

w₁, . . . , w_r |x

. 2.13

From2.12, we can derive the following Barnes-type multipleq-Euler zeta function as follows.

Fors∈C, define ζ_r,q

w₁, w₂, . . . , w_r|s, x

2^r_q ^∞

n1,...,nr0

−1^n1···nrq^n1···nr

n₁w₁n₂w₂· · · n_rw_rxs. 2.14 By2.13and2.14, we obtain the following.

Theorem 2.3. Fork∈Z, w1, w2, . . . , wr ∈C, ζ_r,q

w1, w2, . . . , wr | −k, x Ek,q

w1, w2, . . . , wr|x

. 2.15

Letχbe the primitive Drichlet character with conductorfodd∈N. Then we consider generalized Euler numbers attached toχas follows:

Fχ,qt 2_qf−1

a0−1^aq^aχae^at q^fe^ft1 ^∞

n0

En,χ,qtⁿ

n!, 2.16

where|logqt| < π/f. The numbersEn,χ,q will be called the generalized q-Euler numbers attached toχ. From2.16, note that

Fχ,qt 2_qf−1

a0−1^aq^aχae^at q^fe^ft1 2_q^f−1

a0

−1^aq^aχa^∞

n0

q^nf−1ⁿe^anft 2_q^∞

n0

f−1 a0

−1^anfq^anfχanfe^anft 2_q^∞

n0

−1ⁿqⁿχne^nt∞

n0

E_n,χ,qtⁿ n!.

2.17

Thus,

Ek,χ,q d^k

dt^kFχ,qt

_t0 2_q^∞

n1

−1ⁿqⁿχnn^k, k∈N. 2.18

Therefore, we can define the Dirichlet-typel-function which interpolates at negative integer as follows.

Fors∈C, we definel_qs, χas

lqs, χ 2_q^∞

n1

−1ⁿqⁿχn

n^s . 2.19

By2.18and2.19, we obtain the following.

Theorem 2.4. Fork∈Z,

l_q−k, χ E_k,χ,q. 2.20 From2.1and the definition ofq-Euler numbers, derive

F_qt, x 2_q

qe^t1e^xt∞

n0

E_n,qtⁿ n!

∞ l0

x^l l!t^{l ∞}

m0

_m

n0

E_n,q m

n

x^m−n t^m

m!.

2.21

By2.21it is shown that

En,qx ⁿ

m0

Em,q

n m

x^n−m, n∈Z. 2.22

Forf(=odd)∈N, note that ∞

n0

E_n,qxtⁿ

n! 2_q qe^t1e^xt 2_q 1

q^fe^ft1 f−1

a0

−1^aq^ae^ax/fft 2_q

2_q^f

f−1

a0

−1^aq^a

2_q^fe^ax/fft q^fe^ft1

2_q 2_q^f

f−1

a0

−1^aq^a ∞

n0

E_n,q^f ax

f

fⁿtⁿ n! .

2.23

Thus, we have the distribution relation forq-Euler polynomials as follows.

Theorem 2.5. Forf(=odd)∈N,

En,qx fⁿ2_q 2_q^f

f−1

a0−1^aq^aE_n,q^f ax

f

. 2.24

Fork, n∈Nwithn≡0 (mod 2), it is easy to see that 2_qⁿ⁻¹

l0

−1^l−1q^ll^kqⁿE_k,qn−E_k,q qⁿ

k m0

k m

n^k−mEm,q−Ek,q

qⁿ k−1 m0

k m

Em,qn^k−m qⁿ−1

Ek,q.

2.25

Therefore, we obtain the following.

Theorem 2.6. Fork, n∈Nwithn≡0 (mod 2), 2_qⁿ⁻¹

l0

−1^l−1q^ll^kqⁿ k−1 m0

k m

Em,qn^k−m qⁿ−1

Ek,q. 2.26

3. Witt-type formulae onZpinp-adic number field

In this section, we assume thatq∈Cpwith|1−q|p<1.gis a uniformly differentiable function at a pointa∈Zp, and writeg∈UDZpif the difference quotient

F_gx, y gx−gy

x−y 3.1

has a limitfaasx, y→a, a. Forg∈UDZp, an invariantp-adic,q-integral is defined as I_qg

Zp

gxdμ_qx lim

N→∞

1 p^N

q p^N−1

x0

gxq^x. 3.2

The fermionicp-adic,q-integral is also defined as I_−qg

Zp

gxdμ_−qx lim

N→∞

2_q 1q^pN

p^N−1 x0

gx−1^xq^x 3.3

see4.

From3.3, we have the integral equation as follows:

qI_−qg1 I_−qg 2_qg0, g₁x gx1. 3.4

If we takegx e^tx, then we have Iq

e^tx

Zp

e^xtdμ_−qx 2_q

qe^t1. 3.5

From3.5, we note that ∞

n0

Zp

xⁿdμ_−qxtⁿ

n! 2_q qe^t1 ^∞

n0

E_n,qtⁿ

n!. 3.6

By comparing the coefficient on both sides, we see that

Zp

xⁿdμ_−qx En,q, n∈Z. 3.7

By the same method, we see that

Zp

e^xytdμ_−qy 2_q

qe^t1e^xt∞

n0

En,qxtⁿ

n!. 3.8

Hence, we have the formula of Witt’s type forq-Euler polynomial as follows:

Zp

xyⁿdμ_−qy E_n,qx, n∈Z. 3.9 Forn∈Z, letg_nx gxn. Then we have

qⁿI_−q g_n

−1ⁿ⁻¹I_−qg 2_qⁿ⁻¹

l0

−1^n−1−lq^lgl. 3.10

Ifnis odd positive integer, then we have qⁿI−q

gn

I−qg 2_qⁿ⁻¹

l0

−1^lq^lgl. 3.11 Let χbe the primitive Drichlet character with conduct f odd ∈ Nand let gx χxe^xt. From3.5we derive

I−q

χxe^xt

X

χxe^txdμ_−qx 2_qf−1

a0−1^aq^aχae^at q^fe^ft1 ^∞

n0

En,χ,qtⁿ n!.

3.12

Thus, we have the Witt formula for generalizedq-Euler numbers attached toχas follows:

X

χxxⁿdμ_−qx E_n,χ,q, n≥0. 3.13

4. Higher-orderq-Euler numbers and polynomials

In this section we also assume thatq ∈ Cp with|1−q|p < 1. Now we study on higher-order q-Euler numbers and polynomials and sums of products ofq-Euler numbers. First, we try to consider the multivariate fermionicp-adic,q-integral onZpas follows:

Zp

· · ·

Zp

rtimes

e^{a1x1a2x2···arxrt}e^xtdμ_−q x1

· · ·dμ_−q xr

2^r_q qe^a1t1

qe^a2t1

· · ·

qe^art1e^xt,

4.1

wherea1, a2, . . . , ar∈Zp.

From4.1we consider the multipleq-Euler polynomials as follows:

2^r_q qe^a1t1

qe^a2t1

· · ·

qe^art1e^xt∞

n0

En,q

a1, a2, . . . , ar|xtⁿ

n!. 4.2

In the special casea1, a2, . . . , ar 1,1, . . . ,1,we write En,q

a 1, . . . , ar rtimes

|x

E^rn,qx. 4.3

Forx0, the multipleq-Euler polynomials will be called asq-Euler numbers of orderr. From4.2we note that

En,q

a1, a2, . . . , ar |x

Zp

· · ·

Zp

rtimes

a1x1· · · arxrx_n^r

j1

dμ_−q xj

. 4.4

It is easy to check that

En,q

a1, a2, . . . , ar |x ⁿ

l0

n l

x^n−lEl,q

a1, a2, . . . , ar

, 4.5

whereEn,qa1, a2, . . . , ar En,qa1, a2, . . . , ar |0. Multinomial theorem is well known as fol- lows:

_r

j1

x_{j n}

l1,...,lr≥0 l1···lrn

n l1, . . . , lr

_r

a1

x^l_a^a, 4.6

where

n l1, . . . , lr

n!

l1!l2!· · ·lr!. 4.7

By4.2and4.6we easily see that E^rn,qx ⁿ

m0

l1,...,lr≥0 l1···lrm

n m

m l1, . . . , lr

x^n−m

r j1

Elj,q. 4.8

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