Some Identities on the Generalized -Bernoulli Numbers and Polynomials Associated with -Volkenborn Integrals

Abstract

We give some interesting equation of -adic -integrals on . From those -adic -integrals, we present a systemic study of some families of extended Carlitz type -Bernoulli numbers and polynomials in -adic number field.

1. Introduction

Let be a fixed prime number. Throughout this paper, , , , and will, respectively, denote the ring of -adic rational integer, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of . Let be the set of natural numbers and .

Let be the normalized exponential valuation of with When one talks of -extension, is considered as an indeterminate, a complex number or -adic number If , we normally assume that , and if we normally assume that . We use the notation

(1.1)

The -factorial is defined as

(1.2)

and the Gaussian -binomial coefficient is defined by

(1.3)

(see [1]). Note that

(1.4)

From (1.3), we easily see that

(1.5)

(see [2, 3]). For a fixed positive integer , let

(1.6)

where and (see [1–14]).

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

(1.7)

have a limit as . For , let us begin with the expression

(1.8)

representing a -analogue of the Riemann sums for , (see [1–3, 11–18]). The integral of on is defined as the limit of the sums (if exists). The -adic -integral (= -Volkenborn integral) of ) is defined by

(1.9)

(see [12]). Carlitz's -Bernoull numbers can be defined recursively by and by the rule that

(1.10)

with the usual convention of replacing by , (see [1–13]).

It is well known that

(1.11)

(see [1]), where are called the th Carlitz's -Bernoulli polynomials (see [1, 12, 13]).

Let be the Dirichlet's character with conductor , then the generalized Carlitz's -Bernoulli numbers attached to are defined as follows:

(1.12)

(see [13]). Recently, many authors have studied in the different several areas related to -theory (see [1–13]). In this paper, we present a systemic study of some families of multiple Carlitz's type -Bernoulli numbers and polynomials by using the integral equations of -adic -integrals on . First, we derive some interesting equations of -adic -integrals on . From these equations, we give some interesting formulae for the higher-order Carlitz's type -Bernoulli numbers and polynomials in the -adic number field.

2. On the Generalized Higher-Order -Bernoulli Numbersand Polynomials

In this section, we assume that with . We first consider the -extension of Bernoulli polynomials as follows:

(2.1)

From (2.1), we note that

(2.2)

Note that

(2.3)

where are called the th ordinary Bernoulli polynomials. In the special case, , are called the th -Bernoulli numbers.

By (2.2), we have the following lemma.

Lemma 2.1.

For one has

(2.4)

Now, one considers the -Bernoulli polynomials of order as follows:

(2.5)

By (2.5), one sees that

(2.6)

In the special case, , the sequence is refereed to as the -extension of Bernoulli numbers of order . For , one has

(2.7)

By (2.5) and (2.7), one obtains the following theorem.

Theorem 2.2.

For one has

(2.8)

Let be the primitive Dirichlet's character with conductor , then the generalized -Bernoulli polynomials attached to are defined by

(2.9)

From (2.9), one derives

(2.10)

By (2.9) and (2.10), one can give the generating function for the generalized -Bernoulli polynomials attached to as follows:

(2.11)

From (1.3), (2.10), and (2.11), one notes that

(2.12)

In the special case, , the sequence are called the th generalized -Bernoulli numbers attached to .

Let one consider the higher-order -Bernoulli polynomials attached to as follows:

(2.13)

where are called the th generalized -Bernoulli polynomials of order attaches to .

By (2.13), one sees that

(2.14)

In the special case, , the sequence are called the th generalized -Bernoulli numbers of order attaches to .

By (2.13) and (2.14), one obtains the following theorem.

Theorem 2.3.

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

(2.15)

For , and one introduces the extended higher-order -Bernoulli polynomials as follows:

(2.16)

From (2.16), one notes that

(2.17)

and

(2.18)

In the special case, , are called the th -Bernoulli numbers of order .

By (2.17), one obtains the following theorem.

Theorem 2.4.

For one has

(2.19)

Let be the primitive Dirichlet's character with conductor , then one considers the generalized -Bernoulli polynomials attached to of order as follows:

(2.20)

By (2.20), one sees that

(2.21)

In the special case, , are called the th generalized -Bernoulli numbers attached to of order .

From (2.20) and (2.21), one notes that

(2.22)

By (2.16), it is easy to show that

(2.23)

Thus, one has

(2.24)

From (2.16) and (2.23), one can also derive

(2.25)

It is easy to see that

(2.26)

By (2.23), (2.25), and (2.26), one obtains the following theorem.

Theorem 2.5.

For and one has

(2.27)

Furthermore, one gets

(2.28)

Now, one considers the polynomials of by

(2.29)

By (2.29), one obtains the following theorem.

Theorem 2.6.

For and one has

(2.30)

By using multivariate -adic -integral on , one sees that

(2.31)

Therefore, one obtains the following corollary.

Corollary 2.7.

For and one has

(2.32)

It is easy to show that

(2.33)

From (2.33), one notes that

(2.34)

From the multivariate -adic -integral on , one has

(2.35)

(2.36)

By (2.35) and (2.36), one obtains the following corollary.

Corollary 2.8.

For and one has

(2.37)

Now, one also considers the polynomial of . From the integral equation on , one notes that

(2.38)

By (2.38), one easily gets

(2.39)

Thus, one obtains the following theorem.

Theorem 2.9.

For and one has

(2.40)

From the definition of -adic -integral on , one notes that

(2.41)

Thus, one has

(2.42)

By (2.38), one easily gets

(2.43)

From (2.43), one has

(2.44)

That is,

(2.45)

By (2.38) and (2.43), one easily sees that

(2.46)

and

(2.47)

For , this gives

(2.48)

and

(2.49)

From (2.46) and (2.48), one can derive the recurrence relation for as follows:

(2.50)

where is kronecker symbol.

By (2.46), (2.48), and (2.50), one obtains the following theorem.

Theorem 2.10.

For and one has

(2.51)

Furthermore,

(2.52)

where is kronecker symbol.

From the definition of -adic -integral on , one notes that

(2.53)

By (2.53), one sees that

(2.54)

Note that

(2.55)

where are the th ordinary Bernoulli polynomials.

In the special case, , one gets

(2.56)

It is not difficult to show that

(2.57)

That is,

(2.58)

Let one consider Barnes' type multiple -Bernoulli polynomials. For and one defines Barnes' type multiple -Bernoulli polynomials as follows:

(2.59)

From (2.59), one can easily derive the following equation:

(2.60)

Let , then one has

(2.61)

Therefore, one obtains the following theorem.

Theorem 2.11.

For and one has

(2.62)