Research Article | Open Access

# Multivariate Interpolation Functions of Higher-Order -Euler Numbers and Their Applications

**Academic Editor:**Paul Eloe

#### Abstract

The aim of this paper, firstly, is to construct generating functions of -Euler numbers and polynomials of higher order by applying the fermionic -adic -Volkenborn integral, secondly, to define multivariate -Euler zeta function (Barnes-type Hurwitz -Euler zeta function) and -function which interpolate these numbers and polynomials at negative integers, respectively. We give relation between Barnes-type Hurwitz -Euler zeta function and multivariate -Euler -function. Moreover, complete sums of products of these numbers and polynomials are found. We give some applications related to these numbers and functions as well.

#### 1. Introduction, Definitions, and Notations

Let be a fixed odd prime. Throughout this paper, , , , and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of the algebraic closure of . Let be the normalized exponential valuation of with (cf. [1β28]). When we talk about -extensions, is variously considered as an indeterminate, either a complex , or a -adic number If we assume that If then we assume so that for

For a fixed positive integer with , set where satisfies the condition (cf. [1β28]).

The distribution is given as (cf. [4, 10]).

We say that is a uniformly differentiable function at a point we write if the difference quotient has a limit as Let An invariant -adic -integral is defined by (cf. [4, 5, 10, 29, 30]).

The -extension of is defined by We note that

Classical Euler numbers are defined by means of the following generating function: (cf. [1β3, 5, 8, 9, 15, 16, 18, 19, 20, 23, 28, 30]), where denotes classical Euler numbers. These numbers are interpolated by the Euler zeta function which is defined as follows: (cf. [8, 9, 24, 25, 28]).

-Euler numbers and polynomials have been studied by many mathematicians. These numbers and polynomials are very important in number theory, mathematical analysis and statistics, and the other areas.

In [16], Ozden and Simsek constructed extensions of -Euler numbers and polynomials. In [8], Kim et al. constructed new -Euler numbers and polynomials which are different from Ozden and Simsek [16].

In [31], Kim gave a detailed proof of fermionic -adic -measures on He treated some interesting formulae related -extension of Euler numbers and polynomials. He defined fermionic -adic -measures on as follows: where (cf. [1, 31]).

By using the fermionic -adic -measures, he defined the fermionic -adic -integral on as follows: (cf. [31]).

Observe that can be written symbolically as (cf. [31]).

By using fermionic -adic -integral on , Kim et al. [8] defined the generating function of the -Euler numbers as follows: where denotes -Euler numbers.

Witt's formula of was given by Kim et al. [8]: where and

In [16], Ozden and Simsek defined generating function of -Euler numbers by

In [7, 9], Kim defined --functions and -multiple -functions. He also gave many applications of these functions.

We summarize our paper as follows. In Section 2, we give some fundamental properties of the -Euler numbers and polynomials. We also give some relations related to these numbers and polynomials. By using generating functions of -Euler numbers and polynomials of higher order, we define multivariate -Euler zeta function (Barnes-type Hurwitz -Euler zeta function) and -function which interpolate these numbers and polynomials at negative integers. We also give contour integral representation of these functions. In Section 3, we find relation between and . By using these relations, we obtain distribution relations of the generalized -Euler numbers and polynomials of higher order. In Section 4, we find complete sums of products of these numbers and polynomials. We also give some applications related to these numbers and functions.

#### 2. Some Properties of -Euler Numbers and Polynomials

For with (cf. [8]), where denotes the -Euler number and

Observe that by (2.1) we have From (2.1) and (2.2), we note that where are called Frobenius Euler numbers (cf. [27, 28]).

The -Euler polynomials are also defined by means of the following generating function [8]: where By comparing the coefficients of on both sides of the above equation, we have the following theorem.

Theorem 2.1. *Let be nonnegative
integer. Then **
with the usual convention about
replacing by *
By using (2.5), we have
From (2.3), by applying
Cauchy product and using (2.1), we also obtain
By comparing the coefficients of on both sides of the above equation, we have
(cf. [8, 14]).

By using Theorem 2.1 and [8, equation (3)], we obtain

By using the above equation, we arrive at the following theorem.

Theorem 2.2. *Let be odd. Then *

By simple calculation in (2.3), Ryoo et al. [14] give another proof of Theorem 2.2, which is given as follows: let be odd; By comparing the coefficients of on both sides of the above equation, we have Theorem 2.2.

By substituting , with into (2.3), then we have Thus, Hence, by (2.13), we have By the generating function of -Euler numbers and polynomials and by (2.14), we see that By comparing the coefficients of on both sides of (2.15), we obtain the following alternating sums of powers of consecutive -integers as follows.

Theorem 2.3 (see [14]). *Let Then *

*Remark 2.4. * Proof of
Theorem 2.3 is similar to that of [14]. If we take in (2.16), we have

The above formula is well known in the number theory and its applications.

*Remark 2.5. *Generating
function of the -Euler numbers in this paper is different than that in [29, 31]. It is same as in [8]. Consequently, all these generating functions in [8, 16, 29, 31] produce different-type -Euler numbers.
But we observe that all these generating functions
were obtained by the same fermionic -adic -measures on and the
fermionic -adic -integral on for applications of this integral and measure see
for detail [2, 4, 8, 14β19, 23, 25, 29, 30, 31].

Now, we consider -Euler numbers and polynomials of higher order as follows: where are called -Euler numbers of order We also consider -Euler polynomials of order as follows: where From these generating functions of -Euler numbers and polynomials of higher order, we construct multiple -Euler zeta functions. First, we investigate the properties of generating function of -Euler polynomials of higher order as follows:

By applying Mellin transformation to (2.20), we have After some elementary calculations, we obtain

From (2.22), we define the analytic function which interpolates higher-order -Euler numbers at negative integers as follows.

*Definition 2.6. *For
one defines
is called Barnes-type Hurwitz -Euler zeta function.

*Remark 2.7. *By applying
the th derivative
operator on both sides
of (2.20), we have
By using the above equation, Ryoo et al. [14] and Simsek [23] also define (2.23).

By substituting into (2.23) and using (2.24), after some calculations, we arrive at the following theorem.

Theorem 2.8. *Let Then *

Observe that the function interpolates polynomial at negative integers. By using the complex integral representation of generating function of the polynomials we have where is Hankel's contour along the cut joining the points and on the real axis, which starts from the point at encircles the origin once in the positive (counter-clockwise) direction, and returns to the point at (see for detail [13, 17, 25, 28]). By using (2.26)) and Cauchy-Residue theorem, then we arrive at (2.25).

*Remark 2.9. * is called
Barnes-type -Euler zeta
function; see for detail [14]. is an analytic
function in whole complex -plane. For
If in the above
equation, we have
The function is known as classical Hurwitz-type zeta function which interpolates classical Euler numbers
at negative integers, cf. [28].

Let be Dirichlet's character with conductor The generalized -Euler numbers attached to of higher order are defined by (cf. [8]), where The -Euler numbers attached to of higher order are defined by From (2.30), we obtain By applying the th derivative operator in (2.31), we have By using (2.32), we define Dirichlet-type multiple Euler --function as follows.

*Definition 2.10. *Let

*Remark 2.11. * is an analytic
function in the whole complex -plane. From
the above definition,
For in the above
equation, we have
This function is called Euler -function.

Here, we observe that by applying Mellin
transformation to (2.31), we obtain
This gives us another definition of (2.32).

By substituting into (2.33) and using (2.32), we arrive at the following theorem.

Theorem 2.12. * Let Then *

We note that where are called classical Euler numbers attached to of higher order, cf. [28]. By using (2.26), (2.36), we obtain another proof of (2.37).

#### 3. Relation between and

Substituting where and where and is odd conductor of , into (2.33), we have By substituting (2.23) into the above equation, we arrive at the following theorem.

Theorem 3.1. *
Let be a Dirichlet
character with conductor Then *

By substituting into (3.2), we obtain By using (2.25) and (2.37) in the above equation, we obtain distribution relation of the -Euler numbers attached to of higher order, which is given as follows.

Theorem 3.2. *
The following
holds:
*

#### 4. Multivariate -Adic Fermionic -Integral on Associated with Higher-Order -Euler Numbers

In [14], Ryoo et al. defined -extension of Euler numbers and polynomials of higher order. They studied Barnes-type -Euler zeta functions. They also derived sums of products of -Euler numbers and polynomials by using fermionic -adic -integral. In this section, we assume that with By using (1.4), the -adic fermionic -integral on is defined by From this integral equation, we have (see [1, 2, 4]) where If we take in (4.2), we have (cf. [8]).

Now we are ready to give multivariate -adic fermionic -integral on as follows (see for detail [14]). Let From (4.4), we obtain Witt's formula for -Euler numbers of higher order as follows.

Theorem 4.1 (see [14]). *
Let . Then *

By (4.4), we obtain

Theorem 4.2 (multinomial theorem). *
The
following holds:
**
where are the
multinomial coefficients, which are defined by
**
(cf. [32, 33]).*

Now we give a main theorem of this section, which is called complete sums of products of -Euler polynomials of higher order.

Theorem 4.3. *For positive integers , , one has
**
where is the multinomial coefficient.*

*Proof. * The proof of this theorem is similar to that of [23]. By using Taylor series of into (4.6), and by then we have
By using (4.7) in the above
equation, and after some elementary calculations,
we get
By substituting (2.25) into the above
equation, we arrive at the desired result.

By substituting (2.8) into (4.9), then Theorem 4.3 reduces to the following theorem.

Theorem 4.4. * For positive
integers
one has
*

In (4.10), if we replace by then we obtain the following corollary.

Corollary 4.5. *For
one has
*

*Remark 4.6. * By using (4.5)β(4.7), complete sums of
products of -Euler
polynomials of higher order are also obtained.
Proof of Corollary 4.5 was also given by Ryoo et al. [14], which is given by
In (4.13), if we take , we have
For more detailed information about
complete sums of products of Euler polynomials and Bernoulli
polynomials, see also [11, 14, 20β24, 34, 35].

Let be a Dirichlet character with conductor Then

By using Taylor expansion of and then comparing coefficients of on both sides, we arrive at (cf. [8]).

By (4.16), we have Thus we give Witt-type formula of as follows.

Theorem 4.7. *
Let be a Dirichlet character with conductor and let Then *

By using (3.2), (2.8), we obtain By using (4.7) in the above equation, we have

#### Acknowledgments

The first and the second authors are supported by the research fund of Uludag University Projects no. F-2006/40 and F-2008/31. The third author is supported by the research fund of Akdeniz University. The authors would like to thank the referee for their comments.

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#### Copyright

Copyright Β© 2008 Hacer Ozden 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.