Multivariate Interpolation Functions of Higher-Order -Euler Numbers and Their Applications
Hacer Ozden,1Ismail Naci Cangul,1and Yilmaz Simsek2
Academic Editor: Paul Eloe
Received07 Dec 2007
Accepted22 Jan 2008
Published18 Mar 2008
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. [1–28]). 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. [1–28]).
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. [1–3, 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.
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, 14–19, 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 4.2 (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)–(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, 20–24, 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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