- Research Article
- Open access
- Published:
On Interpolation Functions of the Generalized Twisted
-Euler Polynomials
Journal of Inequalities and Applications volume 2009, Article number: 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 [3–48]. 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 [1–32, 32–65]. 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. [1–32, 32–48, 50, 51, 54–65]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ1_HTML.gif)
Hence, , for any
with
in the present
-adic case.
For a fixed positive integer with
, set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ2_HTML.gif)
where satisfies the condition
. The distribution is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ3_HTML.gif)
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]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ4_HTML.gif)
The fermionic -adic
-measures on
is defined as (cf. [14–16, 18, 22, 28])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ5_HTML.gif)
for . For
, the ferminoic
-adic invariant
-integral on
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ6_HTML.gif)
which has a sense as we see readily that the limit is convergent. For , we note that (cf. [14, 16, 18, 22, 28])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ7_HTML.gif)
From the fermionic invariant integral on , we derive the following integral equation (cf. [14, 35]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ8_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ9_HTML.gif)
Then, we denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ10_HTML.gif)
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])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ11_HTML.gif)
By induction in (1.8), Kim constructed the following useful identity (cf. [14, 28]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ12_HTML.gif)
where . From (2.4), if
is odd, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ13_HTML.gif)
If we replace by
into (2.5), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ14_HTML.gif)
Let . Let
be a Dirichlet's character of conductor
, which
is any multiple of
with
. By substituting
into (2.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ15_HTML.gif)
Remark 2.1.
In complex case, the generating function of the Euler numbers is given by (cf. [28])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ16_HTML.gif)
By using Taylor series of , then we can define the generalized twisted Euler numbers
attached to
as follows (cf. [55]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ17_HTML.gif)
In [8], -Euler numbers were defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ18_HTML.gif)
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]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ19_HTML.gif)
For and
, we have that (cf. [55])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ21_HTML.gif)
where with
.
Let be the generating function of
in complex plane as follows (cf. [55]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ22_HTML.gif)
Let be the Dirichlet's character with conductor
with
. Then the generalized twisted
-Euler polynomials attached to
is given by as follows:
For ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ23_HTML.gif)
where is any multiple of
with
and
.
Then the distribution relation of the generalized twisted -Euler polynomials is given by as follows (cf. [14]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ24_HTML.gif)
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]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ25_HTML.gif)
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]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ26_HTML.gif)
where and
. Note that if
in (3.2), then we see that the twisted
-Euler-zeta function is defined by (cf. [28, 55])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ27_HTML.gif)
For , we know (cf. [28])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ28_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ29_HTML.gif)
By substituting and
into (3.5), then using (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ30_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ31_HTML.gif)
By substituting with
, into (3.7), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ32_HTML.gif)
where .
Thus, we have the following theorem.
Theorem 3.3.
For , let
be the Dirichlet's character with conductor
with
. Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ33_HTML.gif)
Remark 3.4.
If we take in (3.5), then we have (cf. [28, 55])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ36_HTML.gif)
We consider the well-known identity (cf. [44, 65])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ37_HTML.gif)
By using (3.12), we define two-variable twisted Euler -
-function as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ38_HTML.gif)
We will investigate the relations between and
as follows.
Substituting with
into (3.12), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ39_HTML.gif)
Thus we obtain the following theorem.
Theorem 3.7.
For with
, let
be the Dirichlet character with conductor
with
and
with
. Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ40_HTML.gif)
By substituting with
into (3.16) and using (3.4), we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ41_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ42_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ43_HTML.gif)
By substituting , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ44_HTML.gif)
By substituting (3.2), for , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ45_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ46_HTML.gif)
Remark 3.10.
If we take in (3.22), then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ48_HTML.gif)
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 Teichmüller character having conductor
. For an arbitrary character
, we define
, where
, in the sense of the product of characters. Let
. Then
. Hence we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ49_HTML.gif)
where with
.
We denote the subset of
by (cf. [62])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ50_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ51_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ52_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ53_HTML.gif)
Then is analytic for
with
, when
. For
with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ54_HTML.gif)
is analytic for . It readily follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ55_HTML.gif)
is analytic for with
when
. Thus we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ56_HTML.gif)
Let and fixed
with
. Then we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ57_HTML.gif)
If , then
, so
is a multiple of
. Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ58_HTML.gif)
Then we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ59_HTML.gif)
The difference of these equations yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ60_HTML.gif)
Using distribution for -Euler polynomials, we easily see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ61_HTML.gif)
Since , for
, and
, with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ62_HTML.gif)
From (4.5)–(4.14), we can derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ63_HTML.gif)
Therefore we obtain the following theorem.
Theorem 4.1.
Let be a positive integral multiple of
and
with
, and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ64_HTML.gif)
Then is analytic for
, provides
when
. Furthermore, for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ65_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ66_HTML.gif)
for .
From (4.18), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ67_HTML.gif)
By using the definition of , we can express
for all
and
with
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ68_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ69_HTML.gif)
The -adic logarithmic function,
, is the unique function
that satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ70_HTML.gif)
By employing these expansion and some algebraic manipulations, we evaluate the derivative . It follows from the definition of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ71_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ72_HTML.gif)
Since is a root of unity for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ73_HTML.gif)
Thus we have the following theorem.
Theorem 4.2.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_IEq367_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F946569/MediaObjects/13660_2008_Article_2041_Equ74_HTML.gif)
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-analogue of the
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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