Multiple Twisted -Euler Numbers and Polynomials Associated with -Adic -Integrals

Let be a fixed odd prime number. Throughout this paper, , , and are, respectively, the ring of -adic rational integers, the field of -adic rational numbers, and the -adic completion of the algebraic closure of . The -adic absolute value in is normalized so that . When one talks about -extension, is variously considered as an indeterminate, a complex number, or a -adic number . If , one normally assumes that . If , one normally assumes that so that for each . We use the notations

(1.1)

  (cf. [1–14]), for all . For a fixed odd positive integer with , set

(1.2)

where lies in . For any ,

(1.3)

is known to be a distribution on (cf. [1–28]).

We say that is uniformly differentiable function at a point and denote this property by if the difference quotients

(1.4)

have a limit as (cf. [25]).

The -adic -integral of a function was defined as

(1.5)

(1.6)

(cf. [4, 24, 25, 28]), from (1.6), we derive

(1.7)

where . If we take , then we have . From (1.7), we obtain that

(1.8)

In Section 2, we define the multiple twisted -Euler numbers and polynomials on and find Witt's type formula for multiple twisted -Euler numbers. We also have sums of consecutive multiple twisted -Euler numbers. In Section 3, we consider multiple twisted -Euler Zeta functions which interpolate new multiple twisted -Euler polynomials at negative integers and investigate some characterizations of them. In Section 4, we construct the multiple twisted Barnes' type -Euler polynomials and multiple twisted Barnes' type -Euler Zeta functions which interpolate new multiple twisted Barnes' type -Euler polynomials at negative integers. In Section 5, we define multiple twisted Dirichlet's type -Euler numbers and polynomials and give Witt's type formula for them.

In this section, we assume that with . For , by the definition of -adic -integral on , we have

(2.1)

where . If is odd positive integer, we have

(2.2)

Let be the locally constant space, where is the cyclic group of order . For , we denote the locally constant function by

(2.3)

(cf. [5, 7–14, 16, 18]). If we take , then we have

(2.4)

Now we define the twisted -Euler numbers as follows:

(2.5)

We note that by substituting , are the familiar Euler numbers. Over five decades ago, Carlitz defined -extension of Euler numbers (cf. [15]). From (2.4) and (2.5), we note that Witt's type formula for a twisted -Euler number is given by

(2.6)

for each and .

Twisted -Euler polynomials are defined by means of the generating function

(2.7)

where . By using the th iterative fermionic -adic -integral on , we define multiple twisted -Euler number as follows:

(2.8)

Thus we give Witt's type formula for multiple twisted -Euler numbers as follows.

Theorem 2.1.

For each and ,

(2.9)

where

(2.10)

From (2.8) and (2.9), we obtain the following theorem.

Theorem 2.2.

For and ,

(2.11)

From these formulas, we consider multivariate fermionic -adic -integral on as follows:

(2.12)

Then we can define the multiple twisted -Euler polynomials as follows:

(2.13)

From (2.12) and (2.13), we note that

(2.14)

Then by the th differentiation on both sides of (2.14), we obtain the following.

Theorem 2.3.

For each and ,

(2.15)

Note that

(2.16)

Then we see that

(2.17)

From (2.15) and (2.17), we obtain the sums of powers of consecutive -Euler numbers as follows.

Theorem 2.4.

For each and ,

(2.18)

For with and , the multiple twisted -Euler numbers can be considered as follows:

(31)

From (3.1), we note that

(32)

By the th differentiation on both sides of (3.2) at , we obtain that

(33)

From (3.3), we derive multiple twisted -Euler Zeta function as follows:

(34)

for all . We also obtain the following theorem in which multiple twisted -Euler Zeta functions interpolate multiple twisted -Euler polynomials.

Theorem 3.1.

For and ,

(35)

In this section, we consider the generating function of multiple twisted -Euler polynomials:

(41)

We note that

(42)

By the th differentiation on both sides of (4.2) at , we obtain that

(43)

Thus we can consider multiple twisted Hurwitz's type -Euler Zeta function as follows:

(44)

for all and . We note that is analytic function in the whole complex -plane and . We also remark that if and , then is Hurwitz's type -Euler Zeta function (see [7, 27]). The following theorem means that multiple twisted -Euler Zeta functions interpolate multiple twisted -Euler polynomials at negative integers.

Theorem 4.1.

For , , , and ,

(45)

Let us consider

(46)

where and . Then will be called multiple twisted Barnes' type -Euler polynomials. We note that

(47)

By the th differentiation of both sides of (4.6), we obtain the following theorem.

Theorem 4.2.

For each , , , and ,

(48)

where

(49)

From (4.8), we consider multiple twisted Barnes' type -Euler Zeta function defined as follows: for each , , , and ,

(410)

We note that is analytic function in the whole complex -plane. We also see that multiple twisted Barnes' type -Euler Zeta functions interpolate multiple twisted Barnes' type -Euler polynomials at negative integers as follows.

Theorem 4.3.

For each , , , and ,

(411)

Let be a Dirichlet's character with conductor and . If we take , then we have . From (2.2), we derive

(51)

In view of (5.1), we can define twisted Dirichlet's type -Euler numbers as follows:

(52)

(cf. [17, 19, 21, 22]). From (5.1) and (5.2), we can give Witt's type formula for twisted Dirichlet's type -Euler numbers as follows.

Theorem 5.1.

Let be a Dirichlet's character with conductor . For each , , we have

(53)

We note that if , then is the generalized -Euler numbers attached to (see [18, 26]). From (5.2), we also see that

(54)

By (5.2) and (5.4), we obtain that

(55)

From (5.5), we can define the -function as follows:

(56)

for all . We note that is analytic function in the whole complex -plane. From (5.5) and (5.6), we can derive the following result.

Theorem 5.2.

Let be a Dirichlet's character with conductor . For each , , we have

(57)

Now, in view of (5.1), we can define multiple twisted Dirichlet's type -Euler numbers by means of the generating function as follows:

(58)

where . We note that if , then is a multiple generalized -Euler number (see [22]).

By using the same method used in (2.8) and (2.9),

(59)

From (5.9), we can give Witt's type formula for multiple twisted Dirichlet's type -Euler numbers.

Theorem 5.3.

Let be a Dirichlet's character with conductor . For each , , and , we have

(510)

where and

(511)

From (5.10), we also obtain the sums of powers of consecutive multiple twisted Dirichlet's type -Euler numbers as follows.

Theorem 5.4.

Let be a Dirichlet's character with conductor . For each , , and , we have

(512)

Finally, we consider multiple twisted Dirichlet's type -Euler polynomials defined by means of the generating functions as follows:

(513)

where and . From (5.13), we note that

(514)

Clearly, we obtain the following two theorems.

Theorem 5.5.

Let be a Dirichlet's character with conductor . For each , , , and , we have

(515)

where

(516)

Theorem 5.6.

Let be a Dirichlet's character with conductor . For each , , , and , we have

(517)