In this paper we give some symmetric property of the gener- alized twisted q-Euler zeta functions and q-Euler polynomials

http://dx.doi.org/10.14317/jami.2016.107

SYMMETRIC PROPERTIES FOR GENERALIZED TWISTED q-EULER ZETA FUNCTIONS AND q-EULER POLYNOMIALS

N.S. JUNG AND C.S. RYOO^∗

Abstract. In this paper we give some symmetric property of the gener- alized twisted q-Euler zeta functions and q-Euler polynomials.

AMS Mathematics Subject Classification : 11B68, 11S40, 11S80.

Key words and phrases : generalized twisted q-Euler numbers and polyno- mials, generalized twisted q-Euler zeta function, symmetric property, power sum.

1. Introduction

The Euler numbers and polynomials possess many interesting properties in many areas of mathematics and physics. Many mathematicians have studied in the area of various q-extensions of Euler polynomials and numbers (see [1- 11]). Recently, Y. Hu investigated several identities of symmetry for Carlitz’s q-Bernoulli numbers and polynomials in complex field (see [3]). D. Kim et al.

[4] derived some identities of symmetry for Carlitz’s q-Euler numbers and poly- nomials in complex field. J.Y. Kang and C.S. Ryoo studied some identities of symmetry for q-Genocchi polynomials (see [2]). In [1], we obtained some iden- tities of symmetry for Carlitz’s twisted q-Euler zeta function in complex field.

In this paper, we establish some interesting symmetric identities for generalized twisted q-Euler zeta functions and generalized ] twisted q-Euler polynomials in complex field. If we take χ = 1 in all equations of this article, then [1] are the special case of our results. Throughout this paper we use the following nota- tions. ByN we denote the set of natural numbers, Z denotes the ring of rational integers,Q denotes the field of rational numbers, C denotes the set of complex numbers, andZ+=N ∪ {0}. We use the following notation:

[x]q = 1− q^x

1− q (see [1, 2, 3, 4]).

Received September 20, 2015. Revised November 4, 2015. Accepted November 11, 2015.

∗Corresponding author.

⃝ 2016 Korean SIGCAM and KSCAMc . 107

Note that limq→1[x] = x. We assume that q∈ C with |q| < 1. Let r be a positive integer, and let ε be the r-th root of unity. Let χ be Dirichlet’s character with conductor d ∈ N with d ≡ 1(mod2). Then the generalized twisted q- Euler polynomials associated with associated with χ, En,χ,q,ε, are defined by the following generating function

Fχ,q,ε(t, x) = [2]q

∑∞ n=0

(−1)ⁿqⁿεⁿχ(n)e^[x+n]qt =

∑∞ n=0

En,χ,q,ε(x)tⁿ

n! (1.1) and their values at x = 0 are called the generalized twisted q-Euler numbers and denoted En,χ,q,ε.

By (1.1) and Cauchy product, we obtain E_n,χ,q,ε(x) =

∑n l=0

(n l

)

q^lx[x]ⁿ_q^−lE_l,χ,q,ε, (1.2)

with the usual convention about replacing (Eχ,q,ε)ⁿ by En,χ,q,ε. By using (1.1), we note that

d^k

dt^kF_χ,q,ε(t, x) t=0

= [2]_q

∑∞ n=0

χ(n)(−1)ⁿεⁿqⁿ[n + x]^k_q, (k∈ N). (1.3)

By (1.3), we are now ready to define the Hurwitz type of the generalized twisted q-Euler zeta functions.

Definition 1.1. Let s∈ C and x ∈ R with x ̸= 0, −1, −2, . . .. We define ζχ,q,ε(s, x) = [2]q

∑∞ n=1

(−1)ⁿχ(n)εⁿqⁿ

[n + x]^s_q . (1.4)

Note that ζ_χ,q,ε(s, x) is a meromorphic function on C. Relation between ζ_χ,q,ε(s, x) and E_k,χ,q,ε(x) is given by the following theorem.

Theorem 1.2. For k∈ N, we get

ζ_χ,q,ε(−k, x) = Ek,χ,q,ε(x). (1.5) Observe that ζ_χ,q,ε(−k, x) function interpolates Ek,χ,q,ε(x) polynomials at non-negative integers. If χ = 1, then ζ_χ,q,ε(s, x) = ζ_q,ε(s, x) (see [1]).

2. Symmetric property of generalized twisted q-Euler zeta functions In this section, by using the similar method of [1, 2, 3, 4, 9], expect for obvious modifications, we give some symmetric identities for generalized twisted q-Euler polynomials and generalized twisted q-Euler zeta functions. Let w1, w2∈ N with w1≡ 1 (mod 2), w2≡ 1 (mod 2).

Theorem 2.1. Let χ be Dirichlet’s character with conductor d∈ N with d ≡ 1(mod 2) and ε be the r-th root of unity. For w1, w2∈ N with w1≡ 1 (mod 2), w2≡ 1 (mod 2), we obtain

w∑2d−1 i=0

[2]_qw1[w₁]^s_q(−1)ⁱχ(i)ε^w1iq^w1iζ_χ,qw2,ε^w2

(

s, w₁x +w₁ w2

i )

=

w∑₁d−1 j=0

[2]qw2[w2]^s_q(−1)^jχ(j)ε^w2jq^w2jζχ,qw1,εw1

(

s, w2x + w1

w₂j )

.

Proof. Observe that [xy]_q = [x]_qy[y]_q for any x, y ∈ C. In Definition 1.1, we derive next result by substitute w1x +^w_w¹

2i for x in and replace q and ε by q^w2 and ε^w2, respectively.

ζ_χ,qw2,ε^w2(s, w₁x + w1

w₂i) = [2]_qw2

∑∞ n=0

(−1)ⁿχ(n)ε^w2nq^w2n [n + w₁x + ^w_w¹

2i]^s_qw2

= [2]qw2[w2]^s_q

∑∞ n=0

(−1)ⁿχ(n)ε^w2nq^w2n [w1w2x + w1i + w2n]^s_q.

(2.1)

Since for any non-negative integer n and odd positive integer w1, there exist unique non-negative integer r, j such that m = w1r + j with 0≤ j ≤ w1− 1. So, the equation (2.1) can be written as

ζ_χ,qw2,ε^w2(s, w₁x + w1

w₂i)

= [2]_qw2[w₂]^s_q

∑∞ w₁dr+j=0 0≤j≤w1d−1

(−1)^w1dr+jχ(w₁dr + j)ε^w2(w1dr+j)q^w2(w1dr+j) [w1w2dr + w1w2x + w1i + w2j]^s_q

= [2]_qw2[w₂]^s_q

w∑1d−1 j=0

∑∞ r=0

(−1)^jχ(j)ε^w2(w1dr+j)q^w2(w1dr+j) [w1w2(dr + x) + w1i + w2j]^s_q .

(2.2)

In similarly, we obtain

ζ_χ,qw1,ε^w1(s, w₂x + w₂ w1

j) = [2]_qw1

∑∞ n=0

(−1)ⁿχ(n)ε^w1nq^w1n [n + w2x + ^w_w²

1j]^s_qw1

= [2]qw1[w1]^s_q

∑∞ n=0

(−1)ⁿχ(n)ε^w1nq^w1n [w₁w₂x + w₁n + w₂j]^s_q.

(2.3)

Using the method in (2.2), we obtain ζχ,qw1,ζw1(s, w2x + w₂

w1

j)

= [2]qw1[w1]^s_q

∑∞ w2dr+i=0 0≤i≤w2d−1

(−1)^w2dr+iχ(w2dr + i)ε^w1(w2dr+i)q^w1(w2dr+i) [w1w2dr + w1w2x + w1i + w2j]^s_q

= [2]qw1[w1]^s_q

w∑₂d−1 i=0

∑∞ r=0

(−1)ⁱχ(i)ε^w1(w2dr+i)q^w1(w2dr+i) [w₁w₂(dr + x) + w₁i + w₂j]^s_q .

(2.4)

By (2.2) and (2.4), we obtain

w∑₂d−1 i=0

[2]qw1[w1]^s_q(−1)ⁱχ(i)ε^w1iq^w1iζχ,qw2,εw2

(

s, w1x + w1

w₂i )

=

w∑₁d−1 j=0

[2]qw2[w2]^s_q(−1)^jχ(j)ε^w2jq^w2jζχ,qw1,εw1

(

s, w2x + w2

w1

j )

.

(2.5)

 Next, we obtain the symmetric results by using definition and theorem of the generalized twisted q-Euler polynomials.

Theorem 2.2. Let χ be Dirichlet’s character with conductor d∈ N with d ≡ 1(mod 2) and ε be the r-th root of unity. For w1, w2∈ N with w1≡ 1 (mod 2), w2≡ 1 (mod 2), i, j and n be non-negative integer, we obtain

[2]qw1

[w₁]ⁿ_q

w∑₂d−1 i=0

(−1)ⁱχ(i)ε^w1iq^w1iEn,χ,qw2,εw2

(

w1x + w1

w₂i )

=[2]qw2

[w2]ⁿ_q

w∑₁d−1 j=0

(−1)^jχ(j)ε^w2jq^w2jEn,χ,qw1,εw1

(

w2x +w2

w1

j )

.

Proof. By substitute w1x +^w_w¹ⁱ

2 for x in Theorem 1.2 and replace q and ε by q^w2 and ε^w2, respectively, we derive

En,χ,qw2,εw2

(

w1x + w1

w₂i )

= [2]qw2

∑∞ m=0

(−1)^mχ(m)ε^w2mq^w2m [

w1x +w1

w₂i + m ]n

qw2

=[2]_qw2

[w2]ⁿ_q

∑∞ m=0

(−1)^mχ(m)ε^w2mq^w2m[w1w2x + w1i + w2m]ⁿ_q.

(2.6)

Since for any non-negative integer m and odd positive integer w1, there exist unique non-negative integer r, j such that m = w1r + j with 0 ≤ j ≤ w1− 1.

Hence, the equation (2.6) is written as

En,χ,qw2,εw2

(

w1x +w1

w2

i )

= [2]_qw2

[w2]ⁿ_q

∑∞ w1dr+j=0 0≤j≤w1d−1

(−1)^w1dr+jχ(w1dr + j)ε^w2(w1dr+j)q^w2(w1dr+j)

× [w1w2x + w1i + w2(w1dr + j)]ⁿ_q

= [2]qw2

[w2]ⁿ_q

w∑₁d−1 i=0

∑∞ r=0

(−1)^w1dr+jχ(j)ε^w2(w1dr+j)q^w2(w1dr+j)

× [w1w2(x + dr) + w1i + w2j]ⁿ_q.

(2.7)

In similar, we obtain

E_n,χ,qw1,ζ^w1

(

w₂x +w₂ w1

j )

= [2]_qw1

∑∞ m=0

(−1)^mχ(m)ε^w1mq^w1m [

w₂x +w2

w₁j + m ]n

qw1

=[2]qw1

[w1]ⁿ_q

∑∞ m=0

(−1)^mχ(m)ε^w1mq^w1m[w1w2x + w2j + w1m]ⁿ_q,

(2.8)

and

En,χ,qw1,εw1

(

w2x + w2

w₁j )

=[2]qw1

[w₁]ⁿ_q

∑∞ w₂dr+i=0 0≤i≤w2d−1

(−1)^w2dr+iχ(w2dr + i)ε^w1(w2dr+i)q^w1(w2dr+i)

× [w1w2x + w2j + w1(w2dr + i)]ⁿ_q

=[2]_qw1

[w1]ⁿ_q

w∑2d−1 i=0

∑∞ r=0

(−1)^w2dr+iχ(i)ε^w1(w2dr+i)q^w1(w2dr+i)

× [w1w2(x + dr) + w1i + w2j]ⁿ_q.

(2.9)

It follows from the above equation that

[2]qw1

[w1]ⁿ_q

w∑₂d−1 i=0

(−1)ⁱχ(i)ε^w1iq^w1iEn,χ,qw2,εw2

(

w1x +w1

w2

i )

= [2]_qw1

[w1]ⁿ_q [2]_qw2

[w2]ⁿ_q

w∑1d−1 j=0

w∑2d−1 i=0

∑∞ r=0

(−1)^i+jχ(i)χ(j)ε^{w1w2dr+w1i+w2j}

× q^{w1w2dr+w1i+w2j}[w1w2(x + dr) + w1i + w2j)]ⁿ_q

= [2]qw2

[w2]ⁿ_q

w∑₁d−1 j=0

(−1)^jχ(j)ε^w2jq^w2jEn,χ,qw1,εw1

(

w2x + w2

w1

j )

.

(2.10)

From (2.7), (2.8), (2.9) and (2.10), the proof of the Theorem 2.2 is completed. 

By (1.2) and Theorem 2.2, we have the following theorem.

Theorem 2.3. Let i, j and n be non-negative integers. For w1, w2 ∈ N with w1≡ 1 (mod 2), w2≡ 1 (mod 2), we have

[2]qw1

∑n k=0

(n k )

[w1]^k_q[w2]ⁿ_q^−kEn−k,χ,q^w2,εw2(w1x)

×

w∑₂d−1 i=0

(−1)ⁱχ(i)ε^w1iq^(1+n−k)w1i[i]^k_qw1

= [2]qw2

∑n k=0

(n k )

[w1]ⁿ_q^−k[w2]^k_qEn−k,χ,q^w1,εw1(w2x)

×

w∑₁d−1 j=0

(−1)^jχ(j)ε^w2jq^(1+n−k)w2j[j]^k_qw2.

Proof. After some calculations, we have

[2]_qw1

[w1]ⁿ_q

w∑2d−1 i=0

(−1)ⁱχ(i)ε^w1iq^w1iE_n,χ,qw2,ε^w2

(

w₁x + w₁ w2

i )

=[2]qw1

[w1]ⁿ_q

∑n k=0

(n k

) [w1

w2

]k

qw2

En−k,χ,q^w2,εw2(w1x)

×

w∑₂d−1 i=0

(−1)ⁱχ(i)ε^w1iq^(1+n−k)w1i[i]^k_qw1,

(2.11)

and

[2]qw2

[w₂]ⁿ_q

w∑₁d−1 j=0

(−1)^jχ(j)ε^w2jq^w2jE_n,χ,qw2,ε^w2

(

w₂x + w2

w₁j )

= [2]_qw2

[w2]ⁿ_q

∑n k=0

(n k

) [w₁ w2

]k

q^w1

En−k,χ,q^w1,εw1(w2x)

×

w∑1d−1 j=0

(−1)^jχ(j)ε^w2jq^(1+n−k)w2j[j]^k_qw2.

(2.12)

By (2.11), (2.12) and Theorem 2.2, we obtain that

[2]_qw1

∑n k=0

(n k

) 1

[w₁]ⁿq^−k[w₂]^k_qE_n−k,χ,qw2,ε^w2(w₁x)

×

w∑2d−1 i=0

(−1)ⁱχ(i)ε^w1iq^(1+n−k)w1i[i]^k_qw1

= [2]qw2

∑n k=0

(n k

) 1

[w1]^k_q[w2]ⁿq^−k

En−k,χ,q^w1,εw1(w2x)

×

w∑₁d−1 j=0

(−1)^jχ(j)ε^w2jq^(1+n−k)w2j[j]^k_qw2.

Hence, we have above theorem. 

By Theorem 2.3, we have the interesting symmetric identity for generalized twisted q-Euler numbers in complex field.

Corollary 2.4. For w1, w2∈ N with w1≡ 1 (mod 2), w2≡ 1 (mod 2), we have [2]qw1

∑n k=0

(n k )

[w1]^k_q[w2]ⁿ_q^−kEn−k,χ,q^w2,εw2

×

w∑₂d−1 i=0

(−1)ⁱχ(i)ε^w1iq^(1+n−k)w1i[i]^k_qw1

= [2]qw2

∑n k=0

(n k )

[w1]ⁿ_q^−k[w2]^k_qEn−k,χ,q^w1,εw1

×

w∑₁d−1 j=0

(−1)^jχ(j)ε^w2jq^(1+n−k)w2j[j]^k_qw2.

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N.S. Jung received Ph.D. degree from Hannam University. Her research interests are an- alytic number theory and p-adic functional analysis.

Department of Mathematics, Hannam University, Daejeon, 306-791, Korea.

e-mail: jns4235@nate.com

C.S. Ryoo received Ph.D. degree from Kyushu University. His research interests focus on the numerical verification method, scientific computing and p-adic functional analysis.

Department of Mathematics, Hannam University, Daejeon, 306-791, Korea.

e-mail: ryoocs@hnu.kr