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# On Interpolation Functions of the Generalized Twisted -Euler Polynomials

*Journal of Inequalities and Applications*
**volumeÂ 2009**, ArticleÂ number:Â 946569 (2009)

## Abstract

The aim of this paper is to construct -adic twisted two-variable Euler-(,)--functions, which interpolate generalized twisted (,)-Euler polynomials at negative integers. In this paper, we treat twisted (,)-Euler numbers and polynomials associated with -adic invariant integral on . We will construct two-variable twisted (,)-Euler-zeta function and two-variable (,)--function in Complex -plane.

## 1. Introduction

Tsumura and Young treated the interpolation functions of the Bernoulli and Euler polynomials in [1, 2]. Kim and Simsek studied on -adic interpolation functions of these numbers and polynomials [3â€“48]. In [49], Carlitz originally constructed -Bernoulli numbers and polynomials. Many authors studied these numbers and polynomials [4, 28, 38, 41, 50]. After that, twisted -Bernoulli and Euler numbers(polynomials) were studied by several authors [1â€“32, 32â€“65]. In [62], Whashington constructed one-variable -adic--function which interpolates generalized classical Bernoulli numbers at negative integers. Fox introduced the two-variable -adi -functions [53]. Young defined -adic integral representation for the two-variable -adic -functions [64]. Furthermore, Kim constructed the two-variable -adic --function, which is interpolation function of the generalized -Bernoulli polynomials [8]. This function is the -extension of the two-variable -adic -function. Kim constructed -extension of the generalized formula for two-variable of Diamond and Ferrero and Greenberg formula for two-variable -adic -function in the terms of the -adic gamma and log-gamma functions [8]. Kim and Rim introduced twisted -Euler numbers and polynomials associated with basic twisted --functions [28]. Also, Jang et al. investigated the -adic analogue twisted --function, which interpolates generalized twisted -Euler numbers attached to Dirichlet's character [55]. Kim et al. have studied two-variable -adic -functions, which interpolate the generalized Bernoulli polynomials at negative integers. In this paper, we will construct two-variale -adic twisted Euler --functions. This functions interpolation functions of the generalized twisted -Euler polynomials.

Let be a fixed odd prime number. Throughout this paper and will respectively denote the ring of rational integers, the ring of -adic rational integers, the field of -adic rational numbers and the completion of the algebraic closure of . Let be the normalized exponential valuation of such that . If , then . If , we normally assume , so that for . Throughout this paper we use the following notations (cf. [1â€“32, 32â€“48, 50, 51, 54â€“65]):

Hence, , for any with in the present -adic case.

For a fixed positive integer with , set

where satisfies the condition . The distribution is defined by

We say that is uniformly differential function at a point , and we write , if the difference quotients, have a limit as .

For , the -adic invariant -integral on is defined as [4, 18]

The fermionic -adic -measures on is defined as (cf. [14â€“16, 18, 22, 28])

for . For , the ferminoic -adic invariant -integral on is defined as

which has a sense as we see readily that the limit is convergent. For , we note that (cf. [14, 16, 18, 22, 28])

From the fermionic invariant integral on , we derive the following integral equation (cf. [14, 35]):

where .

## 2. Twisted -Euler Numbers and Polynomials

In this section, we will treat some properties of twisted -Euler numbers and polynomials associated with -adic invariant integral on . From now on, we take and with . Let be the space of primitive th root of unity,

Then, we denote

Hence is a -adic locally constant space. For , we denote by defined by , the locally constant function. If we take , then we have (cf. [35])

By induction in (1.8), Kim constructed the following useful identity (cf. [14, 28]):

where . From (2.4), if is odd, then we have

If we replace by into (2.5), we obtain

Let . Let be a Dirichlet's character of conductor , which is any multiple of with . By substituting into (2.6), we have

Remark 2.1.

In complex case, the generating function of the Euler numbers is given by (cf. [28])

By using Taylor series of , then we can define the generalized twisted Euler numbers attached to as follows (cf. [55]):

In [8], -Euler numbers were defined by

where and . In particular, if we take , then . These numbers are called -Euler numbers.

By using iterative method of -adic invariant integral on in the sense of fermionic, we define twisted -Euler numbers as follows (cf. [55]):

For and , we have that (cf. [55])

where with .

Let be the generating function of in complex plane as follows (cf. [55]):

Let be the Dirichlet's character with conductor with . Then the generalized twisted -Euler polynomials attached to is given by as follows:

For ,

where is any multiple of with and .

Then the distribution relation of the generalized twisted -Euler polynomials is given by as follows (cf. [14]):

## 3. Two-Variable Twisted -Euler-Zeta Function and --Function

In this section, we will construct two-variable twisted -Euler-zeta function and two-variable --function in Complex -plane. We assume with .

Firstly, we consider twisted -Euler numbers and polynomials in as follows (cf. [55]):

where and is th root of unity. In particular, if we take , then we have . These numbers are called twisted Euler numbers. By using derivative operator, we have .

From (3.1), we can define Hurwitz-type twisted -Euler-zeta function as follows (cf. [55]):

where and . Note that if in (3.2), then we see that the twisted -Euler-zeta function is defined by (cf. [28, 55])

For , we know (cf. [28])

From now on, we will define the two-variable --functions which interpolates the generalized -Euler polynomials.

Definition 3.1.

Let be the Dirichlet's character with conductor with . For and , we define

By substituting and into (3.5), then using (3.2), we have

Thus, we see the function which interpolates the generalized -Euler polynomials as follows.

Theorem 3.2.

For , let be the Dirichlet's character with conductor with . Then one has

By substituting with , into (3.7), we obtain

where .

Thus, we have the following theorem.

Theorem 3.3.

For , let be the Dirichlet's character with conductor with . Then one has

Remark 3.4.

If we take in (3.5), then we have (cf. [28, 55])

From (3.9) and (3.10), we have the following corollary.

Corollary 3.5.

Let be the Dirichlet's character with conductor with . Then one has

Secondly, we will define two-variable twisted Euler --function as follows.

Definition 3.6.

Let be the Dirichlet's character with conductor with . For and with , we define

We consider the well-known identity (cf. [44, 65])

By using (3.12), we define two-variable twisted Euler --function as follows:

We will investigate the relations between and as follows.

Substituting with into (3.12), we have

Thus we obtain the following theorem.

Theorem 3.7.

For with , let be the Dirichlet character with conductor with and with . Then one has

By substituting with into (3.16) and using (3.4), we can obtain

Thus, we see that the function interpolates generalized -Euler polynomials attached to at negative integer values of as followings.

Theorem 3.8.

For , let be the Dirichlet's character with odd conductor . Then one has

Note that if we take , then Theorem 3.8 reduces to Theorem 3.3.

Let and be integers with and . For , we define partial -Hurwitz type zeta function as follows:

By substituting , we have

By substituting (3.2), for , we get

Equation (3.20) means that the function interpolates polynomials at negative integers.

From (3.16) and (3.20), we have the following theorem.

Theorem 3.9.

For with , let be the Dirichlet's character with conductor with and , is any multiple of . Then one has

Remark 3.10.

If we take in (3.22), then we have

From (2.12), if we take , then we have the following corollary.

Corollary 3.11.

For with , let be the Dirichlet's character with conductor with and , is any multiple of . Then one has

## 4. -Adic Twisted Two-Variable Euler -*L*-Functions

In [62], Washington constructed one-variable -adic--function which interpolates generalized classical Bernoulli numbers negative integers. Kim [22] investigated the -adic analogues of two-variables Euler --function. In this section, we will construct -adic twisted two-variable Euler---functions, which interpolate generalized twisted -Euler polynomials at negative integers. Our notations and methods are essentially due to Kim and Washington (cf. [22, 62]).

We assume that with , so that . Let be an odd prime number. Let denote the TeichmÃ¼ller character having conductor . For an arbitrary character , we define , where , in the sense of the product of characters. Let . Then . Hence we see that

where with .

We denote the subset of by (cf. [62])

Let

be a sequence of power series, each of which converges in a fixed subset such that

(1) as and

(2)for each and , there exists such that

Then for all (cf. [2, 22, 50, 51, 60, 62]).

Let be the Dirichlet's character with conductor with and let be a positive multiple of and .

Now we set

Then is analytic for with , when . For with , we have

is analytic for . It readily follows that

is analytic for with when . Thus we see that

Let and fixed with . Then we have that

If , then , so is a multiple of . Therefore, we have

Then we note that

The difference of these equations yields

Using distribution for -Euler polynomials, we easily see that

Since , for , and , with , we have

From (4.5)â€“(4.14), we can derive that

Therefore we obtain the following theorem.

Theorem 4.1.

Let be a positive integral multiple of and with , and let

Then is analytic for , provides when . Furthermore, for each , we have

Thus we note that for all , where is twisted -adic Euler --function, (cf. [15, 22]).

We now generalized to two-variable -adic Euler --function, which is first defined by the interpolation function

for .

From (4.18), we have that

By using the definition of , we can express for all and with as follows:

We know that is analytic for , when . The value of is the coefficients of in the expansion of at . Using the Taylor expansion at , we see that

The -adic logarithmic function, , is the unique function that satisfies

By employing these expansion and some algebraic manipulations, we evaluate the derivative . It follows from the definition of that

Thus, we have

Since is a root of unity for , we have

Thus we have the following theorem.

Theorem 4.2.

Let be a primitive Dirichlet's character with odd conductor and let be a odd positive integral multiple of and . Then for any with , one has

## References

Tsumura H:

**On a****-adic interpolation of the generalized Euler numbers and its applications.***Tokyo Journal of Mathematics*1987,**10**(2):281â€“293. 10.3836/tjm/1270134514Young PT:

**Congruences for Bernoulli, Euler, and Stirling numbers.***Journal of Number Theory*1999,**78**(2):204â€“227. 10.1006/jnth.1999.2401Kim T:

**On a**-**analogue of the****-adic log gamma functions and related integrals.***Journal of Number Theory*1999,**76**(2):320â€“329. 10.1006/jnth.1999.2373Kim T:

**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288â€“299.Kim T:

**On Euler-Barnes multiple zeta functions.***Russian Journal of Mathematical Physics*2003,**10**(3):261â€“267.Kim T:

**-Riemann zeta function.***International Journal of Mathematics and Mathematical Sciences*2004,**2004**(12):599â€“605. 10.1155/S0161171204307180Kim T:

**Analytic continuation of multiple****-zeta functions and their values at negative integers.***Russian Journal of Mathematical Physics*2004,**11**(1):71â€“76.Kim T:

**Power series and asymptotic series associated with the**-analog of the two-variable -adic**-function.***Russian Journal of Mathematical Physics*2005,**12**(2):186â€“196.Kim T:

**-generalized Euler numbers and polynomials.***Russian Journal of Mathematical Physics*2006,**13**(3):293â€“298. 10.1134/S1061920806030058Kim T:

**A new approach to**-adic -**-functions.***Advanced Studies in Contemporary Mathematics*2006,**12**(1):61â€“72.Kim T:

**A note on**-adic invariant integral in the rings of**-adic integers.***Advanced Studies in Contemporary Mathematics*2006,**13**(1):95â€“99.Kim T:

**Multiple**-adic**-function.***Russian Journal of Mathematical Physics*2006,**13**(2):151â€“157. 10.1134/S1061920806020038Kim T:

**-extension of the Euler formula and trigonometric functions.***Russian Journal of Mathematical Physics*2007,**14**(3):275â€“278. 10.1134/S1061920807030041Kim T:

**On the analogs of Euler numbers and polynomials associated with**-adic -integral on at .*Journal of Mathematical Analysis and Applications*2007,**331**(2):779â€“792. 10.1016/j.jmaa.2006.09.027Kim T: On -adic -

**-functions and sums of powers.***Journal of Mathematical Analysis and Applications*2007,**329**(2):1472â€“1481. 10.1016/j.jmaa.2006.07.071Kim T:

**On the****-extension of Euler and Genocchi numbers.***Journal of Mathematical Analysis and Applications*2007,**326**(2):1458â€“1465. 10.1016/j.jmaa.2006.03.037Kim T:

**A note on**-adic -integral on associated with**-Euler numbers.***Advanced Studies in Contemporary Mathematics*2007,**15**(2):133â€“137.Kim T:

**-Euler numbers and polynomials associated with**-adic**-integrals.***Journal of Nonlinear Mathematical Physics*2007,**14**(1):15â€“27. 10.2991/jnmp.2007.14.1.3Kim T:

**Euler numbers and polynomials associated with zeta functions.***Abstract and Applied Analysis*2008,**2008:**-11.Kim T:

**-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.***Russian Journal of Mathematical Physics*2008,**15**(1):51â€“57.Kim T:

**Note on the Euler numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**17**(2):109â€“115.Kim T:

**On**-adic interpolating function for**-Euler numbers and its derivatives.***Journal of Mathematical Analysis and Applications*2008,**339**(1):598â€“608. 10.1016/j.jmaa.2007.07.027Kim T:

**A note on****-Euler numbers and polyomials.***Advanced Studies in Contemporary Mathematics*2008,**16**(2):161â€“170.Kim T:

**The modified****-Euler numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**16**(2):161â€“170.Kim T:

**On a**-adic interpolation function for the**-adic interpolation function for the -extension of the generalized Bernoulli polynomials and its derivative.***Discrete Mathematics*2009,**309**(6):1593â€“1602. 10.1016/j.disc.2008.03.001Kim T:

**Note on the Euler -zeta functions.***Journal of Number Theory*. In press Journal of Number Theory. In pressKim T, Choi JY, Sug JY:

**Extended**-Euler numbers and polynomials associated with fermionic -adic -integral on .*Russian Journal of Mathematical Physics*2007,**14**(2):160â€“163. 10.1134/S1061920807020045Kim T, Rim S-H:

**On the twisted**-Euler numbers and polynomials associated with basic -**-functions.***Journal of Mathematical Analysis and Applications*2007,**336**(1):738â€“744. 10.1016/j.jmaa.2007.03.035Ozden H, Cangul IN, Simsek Y:

**Remarks on sum of products of**(,)**-twisted Euler polynomials and numbers.***Journal of Inequalities and Applications*2008,**2008:**-8.Ozden H, Simsek Y, Cangul IN:

**Euler polynomials associated with**-adic**-Euler measure.***General Mathematics*2007,**15**(2):24â€“37.Ozden H, Simsek Y:

**A new extension of****-Euler numbers and polynomials related to their interpolation functions.***Applied Mathematics Letters*2008,**21**(9):934â€“939. 10.1016/j.aml.2007.10.005Cangul IN, Ozden H, Simsek Y:

**Generating functions of the**(, )**extension of twisted Euler polynomials and numbers.***Acta Mathematica Hungarica*2008,**120**(3):281â€“299. 10.1007/s10474-008-7139-1Ozden H, Cangul IN, Simsek Y:

**Multivariate interpolation functions of higher-order****-Euler numbers and their applications.***Abstract and Applied Analysis*2008,**2008:**-16.Simsek Y, Yurekli O, Kurt V:

**On interpolation functions of the twisted generalized Frobenius-Euler numbers.***Advanced Studies in Contemporary Mathematics*2007,**15**(2):187â€“194.Rim S-H, Kim T:

**A note on**-Euler numbers associated with the basic**-zeta function.***Applied Mathematics Letters*2007,**20**(4):366â€“369. 10.1016/j.aml.2006.04.019Simsek Y:

**Theorems on twisted****-function and twisted Bernoulli numbers.***Advanced Studies in Contemporary Mathematics*2005,**11**(2):205â€“218.Simsek Y: -analogue of twisted -series and

**-twisted Euler numbers.***Journal of Number Theory*2005,**110**(2):267â€“278. 10.1016/j.jnt.2004.07.003Simsek Y:

**On**-adic twisted -**-functions related to generalized twisted Bernoulli numbers.***Russian Journal of Mathematical Physics*2006,**13**(3):340â€“348. 10.1134/S1061920806030095Simsek Y:

**Hardy character sums related to Eisenstein series and theta functions.***Advanced Studies in Contemporary Mathematics*2006,**12**(1):39â€“53.Simsek Y:

**Remarks on reciprocity laws of the Dedekind and Hardy sums.***Advanced Studies in Contemporary Mathematics*2006,**12**(2):237â€“246.Simsek Y:

**Twisted**(,)-Bernoulli numbers and polynomials related to twisted (,)-zeta function and**-function.***Journal of Mathematical Analysis and Applications*2006,**324**(2):790â€“804. 10.1016/j.jmaa.2005.12.057Simsek Y:

**On twisted**-Hurwitz zeta function and -two-variable**-function.***Applied Mathematics and Computation*2007,**187**(1):466â€“473. 10.1016/j.amc.2006.08.146Simsek Y:

**The behavior of the twisted**-adic (,)--functions at**-functions at .***Journal of the Korean Mathematical Society*2007,**44**(4):915â€“929. 10.4134/JKMS.2007.44.4.915Simsek Y:

**Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions.***Advanced Studies in Contemporary Mathematics*2008,**16**(2):251â€“278.Simsek Y, Kim D, Rim S-H:

**On the two-variable Dirichlet**-**-series.***Advanced Studies in Contemporary Mathematics*2005,**10**(2):131â€“142.Simsek Y, Mehmet A:

**Remarks on Dedekind eta function, theta functions and Eisenstein series under the Hecke operators.***Advanced Studies in Contemporary Mathematics*2005,**10**(1):15â€“24.Ozden H, Cangul IN, Simsek Y:

**On the behavior of two variable twisted -adic Euler --functions.***Nonlinear Analysis*. In press Nonlinear Analysis. In pressSimsek Y, Yang S:

**Transformation of four Titchmarsh-type infinite integrals and generalized Dedekind sums associated with Lambert series.***Advanced Studies in Contemporary Mathematics*2004,**9**(2):195â€“202.Carlitz L:

**-Bernoulli and Eulerian numbers.***Transactions of the American Mathematical Society*1954,**76:**332â€“350.Cenkci M, Can M:

**Some results on****-analogue of the Lerch zeta function.***Advanced Studies in Contemporary Mathematics*2006,**12**(2):213â€“223.Cenkci M, Simsek Y, Kurt V:

**Further remarks on multiple**-adic -**-function of two variables.***Advanced Studies in Contemporary Mathematics*2007,**14**(1):49â€“68.DÄ…browski A:

**A note on**-adic -**-functions.***Journal of Number Theory*1997,**64**(1):100â€“103. 10.1006/jnth.1997.2107Fox GJ:

**A**-adic**-function of two variables.***L'Enseignement MathÃ©matique, IIe SÃ©rie*2000,**46**(3â€“4):225â€“278.Jang L-C, Kim S-D, Park D-W, Ro Y-S:

**A note on Euler number and polynomials.***Journal of Inequalities and Applications*2006,**2006:**-5.Jang L-C, Kurt V, Simsek Y, Rim SH: -analogue of the -adic twisted -function.

*Journal of Concrete and Applicable Mathematics*2008,**6**(2):169â€“176.Koblitz N:

**On Carlitz's****-Bernoulli numbers.***Journal of Number Theory*1982,**14**(3):332â€“339. 10.1016/0022-314X(82)90068-3Park KH, Kim Y-H:

**On some arithmetical properties of the Genocchi numbers and polynomials.***Advances in Difference Equations*2008, -14.Rim S-H, Park KH, Moon EJ:

**On Genocchi numbers and polynomials.***Abstract and Applied Analysis*2008,**2008:**-7.Schikhof WH:

*Ultrametric Calculus: An Introduction to p-Adic Analysis, Cambridge Studies in Advanced Mathematics*.*Volume 4*. Cambridge University Press, Cambridge, UK; 1984:viii+306.Shiratani K, Yamamoto S:

**On a****-adic interpolation function for the Euler numbers and its derivatives.***Memoirs of the Faculty of Science, Kyushu University. Series A*1985,**39**(1):113â€“125. 10.2206/kyushumfs.39.113Srivastava HM, Kim T, Simsek Y: -Bernoulli numbers and polynomials associated with multiple -zeta functions and basic

**-series.***Russian Journal of Mathematical Physics*2005,**12**(2):241â€“268.Washington LC:

*Introduction to Cyclotomic Fields, Graduate Texts in Mathematics*.*Volume 83*. 2nd edition. Springer, New York, NY, USA; 1997:xiv+487.Woodcock CF:

**Special****-adic analytic functions and Fourier transforms.***Journal of Number Theory*1996,**60**(2):393â€“408. 10.1006/jnth.1996.0130Young PT:

**On the behavior of some two-variable**-adic**-functions.***Journal of Number Theory*2003,**98**(1):67â€“88. 10.1016/S0022-314X(02)00031-8Zhao J:

**Multiple**-zeta functions and multiple -**polylogarithms.***Ramanujan Journal*2007,**14**(2):189â€“221. 10.1007/s11139-007-9025-9

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Park, K.H. On Interpolation Functions of the Generalized Twisted -Euler Polynomials.
*J Inequal Appl* **2009**, 946569 (2009). https://doi.org/10.1155/2009/946569

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DOI: https://doi.org/10.1155/2009/946569