Báo cáo hóa học: " Research Article Some Relationships between the Analogs of Euler Numbers and Polynomials" doc

Next we construct twisted Euler zeta function, twisted Hurwitz zeta func-tion, twisted Dirichletl-Euler numbers and twisted Euler polynomials at non-positive integers, respectively.. The

Volume 2007, Article ID 86052, 22 pages

doi:10.1155/2007/86052

Research Article

Some Relationships between the Analogs of

Euler Numbers and Polynomials

C S Ryoo, T Kim, and Lee-Chae Jang

Received 5 June 2007; Revised 28 July 2007; Accepted 26 August 2007

Recommended by Narendra K Govil

We construct new twisted Euler polynomials and numbers We also study the generatingfunctions of the twisted Euler numbers and polynomials associated with their interpola-tion functions Next we construct twisted Euler zeta function, twisted Hurwitz zeta func-tion, twisted Dirichletl-Euler numbers and twisted Euler polynomials at non-positive

integers, respectively Furthermore, we find distribution relations of generalized twistedEuler numbers and polynomials By numerical experiments, we demonstrate a remark-ably regular structure of the complex roots of the twistedq-Euler polynomials Finally,

we give a table for the solutions of the twistedq-Euler polynomials.

Copyright © 2007 C S Ryoo et al This is an open access article distributed under theCreative 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

rational integers,Qdenotes the field of rational numbers,Qpdenotes the field ofp-adic

rational numbers,Cdenotes the complex numbers field, andCpdenotes the completion

of algebraic closure ofQp Letν p be the normalized exponential valuation ofCp with

| p | p = p − ν p(p) = p −1 When one talks ofq-extension, q is considered in many ways such

as an indeterminate, a complex numberq ∈ C, or p-adic number q ∈ C p Ifq ∈ C, onenormally assumes that| q | < 1 If q ∈ C p, we normally assume that| q −1| p < p −1/(p −1)sothatq x =exp(x log q), for | x | p ≤1

Hence, limq →1[x] = x for any x with | x | p ≤1 in the presentp-adic case Let d be a fixed

integer and letp be a fixed prime number For any positive integer N, we set

(see [1,2,6–18]) Forq ∈[0, 1], certainq-deformed bosonic operators may be introduced

which generalize the undeformed bosonic ones (correspondingq =1); see [1,2,6–18].Forg ∈ UD(Z p)

(see [6–18] for details)

We assume thatq ∈ Cwith|1− q | p < 1 Using definition, we note that I1( 1)= I1(g) +

=1}is the cyclic group of order p m Forw ∈ T p, we denote by

φ w:Zp → C pthe locally constant functionx −→ w x If we take f (x) = φ w(x)e tx, then we

easily see that

E nto the so-called Frobenius-Euler numbersH n(u) belonging to an algebraic number u

with| u | > 1 Let u be an algebraic number For u ∈ Cwith| u | > 1, the Frobenius-Euler

numbersH n(u) belonging to u are defined by the generating function

Now, we consider the caseq ∈(−1, 0) corresponding toq-deformed fermionic certain

and annihilation operators and the literature given therein [6–18] The expression for the

I q(g) remains the same, so it is tempting to consider the limit q → −1 That is,

Lemma 1.1 For g ∈ UD(Z p ), one has

Theorem 1.2 For g ∈ UD(Z p ), n ∈ N , one has

Forw ∈ T p, we introduce the twisted Euler polynomialsE n,w(z) Twisted Euler

poly-nomialsE n,w(z) are defined by means of the generating function



z n − k E k,w (1.27)

Letχ be the Dirichlet character with conductor f ( =odd)∈ N Ryoo et al [16] studiedthe generalized Euler numbers and polynomials The generalized Euler numbers associ-ated withχ, E n,χ, were defined by means of the generating function

Substitutingg(x) = χ(x)φ w(x)e txinto (1.21), then the generalized twisted Euler numbers

E n,χ,ware defined by means of the generating functions

We have the following remark

Remark 1.4 Note that

Comparing the coefficients tnon both sides of the above equation, we arrive at (1.35)

2 Twisted zeta function

In this section, we introduce the twisted Euler zeta function and twisted Hurwitz-Eulerzeta function We derive a new twisted Hurwitz-typel-function which interpolates the

generalized Euler polynomialsE n,χ,w(x) We give the relation between twisted Euler

num-bers and twistedl-functions at nonpositive integers Let χ be the Dirichlet character with

conductor f ( =odd)∈ N We set

Theorem 2.1 For k ∈ N , one has

Definition 2.2 For s ∈ C, define the Dirichlet-typel-series related to twisted Euler

Therefore, we have the following theorem.

Theorem 2.4 For k ∈ N , one has

E k,w(z) =2

∞

n =0(−1)n w n(n + z) k (k ∈ N). (2.10)Thus the twisted Hurwitz-Euler zeta function is defined as follows

Definition 2.5 Let s ∈ C Then

Theorem 2.6 For k ∈ N , one obtains

ζ E,w(− k, z) = E k,w(z). (2.12)Let us define two-variable twisted Euler numbers attached toχ as follows By (1.33),

Theorem 2.7 For k ∈ N , one has

E k,χ,w(z) =2

∞

n =0(−1)n χ(n)w n(n + z) k (k ∈ N). (2.15)Hence we define two-variable twistedl-series as follows.

Definition 2.8 For s ∈ C Then

The relation betweenl w(− k, χ | z) and E k,χ,w(z) is given by the following theorem.

Theorem 2.9 For k ∈ N , one obtains

s, a + z f



s, a + z f



s, a f



− n, a + z f



a + z f

n

k =0

n k

n

k =0

n k

n − k j

Theorem 2.11 For n ∈ Z+, one obtains

n − k j



f k z n − k − j E k,w f S w,χ(j). (2.25)

3 Twistedq-Euler zeta function and twisted q-analog Dirichlet l-function

Our primary goal of this section is to define generating functions of the twistedq-Euler

numbers and polynomials Using these functions, twistedzeta function and twisted l-functions are defined These functions interpolate twisted q-Euler numbers and gener-

q-alized twistedq-Euler numbers, respectively Now, we introduce the generating functions

F q(t) and F q(x, t) Ryoo et al [15] treated the analog of Euler numbers, which is called

q-Euler numbers in this paper Using p-adic q-integral, we defined the q-Euler numbers



(−1)l 1

1 +q l+1, (3.2)where

F q(t) =[2]q

∞

n =0(−1)n q n e[n] q t (3.4)Note that

Similarly, the generating functionF q(t, z) of the q-Euler polynomials E n,q(z) is defined

Now, we introduce twistedq-Euler numbers E n,q,w Forw ∈ T p, we define twistedq-Euler



(−1)l 1

1 +wq l+1 (3.9)Note that

Definition 3.1 Let s ∈ C Then

Note thatζ E,w(s) is a meromorphic function onC The relation betweenζ q,w(s) and

E k,q,wis given by the following theorem

Theorem 3.2 For k ∈ N , one has

l =0

n l



q lz[z] n − l q

where the symbolE k,q,wis interpreted to mean thatE k

q,wmust be replaced byE k,q,w Using(3.17), we have



(−1)l q lz 1

1 +wq l+1 (3.19)

Using the above equation, we are now ready to define the twisted Hurwitzq-Euler zeta

Note thatζ E,w(s, z) is a meromorphic function onC The relation betweenζ q,w(s, z)

andE k,q,w(z) is given by the following theorem.

Theorem 3.4 For k ∈ N , one has

ζ q,w(− k, z) = E k,q,w(z). (3.24)Observe thatζ q,w(− k, z) function interpolates E k,q,w(z) numbers at nonnegative inte-

gers

4 Distribution and structure of the zeros

In this section, we investigate the zeros of the twistedq-Euler polynomials E n,q,w(z) by

using computer Let w = e2πi/N inC We plot the zeros of E n,q,w(x), x ∈ C, forN =1,

q =1/2 (see Figures4.1,4.2,4.3, and4.4)

4 3 2 1 0 1

Re(z)

2 1 0 1 2 3

Re(z)

2 1 0 1 2 3

4 3 2 1 0 1

Re(z)

2 1 0 1 2 3

Re(z)

2 1 0 1 2 3

4 3 2 1 0 1

Re(z)

2 1 0 1 2 3

Re(z)

2 1 0 1 2 3

4 3 2 1 0 1

Re(z)

2 1 0 1 2 3

Re(z)

2 1 0 1 2 3

Figure 4.8 Zeros ofE40,q,w(x).

We will consider the more general open problem In general, how many roots does

E n,q,w(x) have? Prove or disprove: E n,q(x) has n distinct solutions Find the numbers of

complex zerosC E (x)ofE n,q,w(x), Im(x) =0 Prove or give a counterexample

4 3 2 1 0 1

Re(z)

2 1 0 1 2 3

Re(z)

2 1 0 1 2 3

Figure 4.10 Zeros ofE20,q,w(x).

Conjecture Since n is the degree of the polynomial E n,q,w(x), the number of real zeros

R E (x)lying on the real plane Im(x) =0 is thenR E (x) = n − C E (x), whereC E (x)

4 3 2 1 0 1

Re(z)

2 1 0 1 2 3

Re(z)

2 1 0 1 2 3

Table 4.1 Numbers of real and complex zeros ofE n,q,w(x).

−0.368723 + 0.86325i, 0.603353 −1.50045i, 0.603353 + 1.50045i

employing numerical method in the field of research of theE n,q,w(x) to appear in

mathe-matics and physics The reader may refer to [11,14–16] for the details We calculated anapproximate solution satisfyingE n,q,w(x), N =2, 4,q =1/2, x ∈ C The results are given

in Tables4.2and4.3

5 Further remarks and observations

Using p-adic q-fermionic integral, Rim and Kim [13] studied explicit p-adic

expan-sion for alternating sums of powers In the recent paper [10], Kim and Rim constructed(h, q)-extensions of the twisted Euler numbers and polynomials They also defined (h, q)-

generalizations of the twisted zeta function andL-series These numbers and polynomials

are considered as the (h, extensions of their previous results However, these (h,

q)-Euler numbers and generating functions do not seem to be natural extension of q)-Eulernumbers and polynomials By this reason, we consider the naturalq-extension of Euler

numbers and polynomials In this paper, we include the numerical computations for our

Table 4.3 Approximate solutions ofE n,q,w(x) =0,w = e2πi/4.

1.06026 + 0.618831i

0.164523 + 0.577386i, 1.27487 + 0.650419i

−0.160255 −0.9957i, 0.339376 + 0.642424i, 1.46114 + 0.672881i

twistedq-Euler numbers and polynomials and the Euler numbers and polynomials which

are treated in this paper In [9], many interesting integral equations related to fermionic

p-adic integrals onZpare known We proceed by first constructing generating functions

of the twistedq-Euler polynomials and numbers Then, by applying Mellin

transforma-tion to these generating functransforma-tions, integral representatransforma-tions of the twistedq-Euler zeta

function (andl-functions) are obtained, which interpolate the (generalized) twisted

q-Euler numbers at nonpositive integers

References

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congru-ences forq-twisted and q-generalized twisted Euler numbers,” Advanced Studies in rary Mathematics, vol 9, no 2, pp 203–216, 2004.

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C S Ryoo: Department of Mathematics, Hannam University, Daejeon 306-791, South Korea

T Kim: School of Electronic Engineering and Computer Science, Kyungpook National University, Taegu 702-701, South Korea

Email address:tkim@knu.ac.kr

Lee-Chae Jang: Department of Mathematics and Computer Sciences, KonKuk University,

Chungju 308-701, South Korea

Email addresses:leechae.jang@kku.ac.kr ; leechae-jang@hanmail.net

generalizations of the twisted... zeta function and< i>L-series These numbers and polynomials

are considered as the (h, extensions of their previous results However, these (h,

q) -Euler numbers and generating... 3

Table 4.1 Numbers of real and complex zeros of< /small>E n,q,w(x).

−0.368723