Research Article | Open Access

Volume 2010 |Article ID 860280 | https://doi.org/10.1155/2010/860280

Seog-Hoon Rim, Jeong-Hee Jin, Eun-Jung Moon, Sun-Jung Lee, "Some Identities on the -Genocchi Polynomials of Higher-Order and -Stirling Numbers by the Fermionic -Adic Integral on ", International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 860280, 14 pages, 2010. https://doi.org/10.1155/2010/860280

# Some Identities on the 𝑞 -Genocchi Polynomials of Higher-Order and 𝑞 -Stirling Numbers by the Fermionic 𝑝 -Adic Integral on ℤ 𝑝

Accepted08 Nov 2010
Published29 Dec 2010

#### Abstract

A systemic study of some families of -Genocchi numbers and families of polynomials of Nörlund type is presented by using the multivariate fermionic -adic integral on . The study of these higher-order -Genocchi numbers and polynomials yields an interesting -analog of identities for Stirling numbers.

#### 1. Introduction

Let be a fixed odd prime number. Throughout this paper, , and 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 , respectively. Let be the set of natural numbers and . Let be the normalized exponential valuation of with .

When one talks of -extension, is variously considered as an indeterminate, a complex , or a -adic number . If , then one normally assumes . If , then we assume . In this paper, we use the following notation: see . Hence for all .

The -factorial is defined as , and the Gaussian binomial coefficient is defined by the standard rule (see [7, 9]). Note that . It readily follows from (1.2) that (see [4, 7]).

The -binomial formulas are known,

We say that is uniformly differentiable function at a point , and we write , if the difference quotients such that have a limit as . For , the -deformed fermionic -adic integral is defined as (see [7, 9]). Note that For , write . Then, we have Using (1.7), we can readily derive the Genocchi polynomials, , namely, (see ). Note that are referred to as the th Genocchi numbers. Let us now introduce the Genocchi polynomials of Nörlund type as follows: (see [7, 9]). In the special case , and are referred to as the Genocchi numbers of Nörlund type. Let be the shift operator. Then, the -difference operator is defined as (see [4, 7, 9]). It follows from (1.11) that where (see [5, 6, 10]). The -Stirling number of the second kind (as defined by Carlitz) is given by (see [7, 10]). By (1.12) and (1.13), we see that (see [6, 10]).

In this paper, the -extensions of (1.9) are considered in several ways. Using these -extensions, we derive some interesting identities and relations for Genocchi polynomials and numbers of Nörlund type. The purpose of this paper is to present a systemic study of some families -Genocchi numbers and polynomials of Nörlund type by using the multivariate fermionic -adic integral on .

#### 2.  -Extensions of Genocchi Numbers and Polynomials of Nörlund Type

In this section, we assume that with . We first consider the -extensions of (1.8) given by the rule

Thus, we obtain the following lemma.

Lemma 2.1. If , then

By (1.14), Thus, we have and we obtain the following theorem.

Theorem 2.2. If , then where stand for the nth Genocchi numbers.

Consider a -extensoin of (1.9) such that and Let . Then, In the special case , the numbers are referred to as -extension of the Genocchi numbers of order . In the sense of the -extension in (1.10), consider the -extension of Genocchi polynomials of Nörlund type given by By (2.8), and . Therefore, we obtain the following theorem.

Theorem 2.3. For , and, , write Then,

The numbers are referred to as the -extension of Genocchi numbers of Nörlund type. For and , introduce the extended higher-order -Genocchi polynomials as follows: Then, Let . Then, we can readily see that Therefore, we obtain the following theorem.

Theorem 2.4. For and , let Then,

Let us now define the extended higher-order Nörlund type -Genocchi polynomials as follows: By (2.16), Let . Then, we have where, . Therefore, we obtain the following theorem.

Theorem 2.5. For , , and , write Then, where, .

For , It can readily be seen that By (2.23), . As is known, It follows from (2.24) that By (2.25), A simple manipulation shows that By (2.27), .

Therefore, we obtain the following proposition.

Proposition 2.6. For , and , the following equations hold. Moreover, .

By (2.21), Hence, For , . It also follows from (2.26) that

The Stirling numbers of the first kind are defined as (see[6, 9]), It can readily be seen that By (2.33) and (2.34), Formulas (2.22) and (2.35) imply the following assertion.

Proposition 2.7. For and ,

The generalized Genocchi numbers and polynomials of Nörlund type are defined by and . We can now also define a -extension of (2.37) as follows. For and , write

and . Thus, Another -extension of Nörlund type generalized Genocchi numbers and polynomials is also of interest, namely, and . By (2.40),

#### 3. Further Remarks

For , consider the following polynomials and : Then, Let and let . Then, Consider the following polynomials: A simple calculation of the fermionic -adic invariant integral on show that By (3.5), . It can readily be proved that By (3.6), . Using (2.24), we can also prove that Thus, . For , we have , where is the Kronecker delta.

It is easy to see that . By (3.4), In particular, if , then for .

Recently, Kim has studied -adic fermionic integral on connected with the problems of mathematical physics (see [6, 10, 11]), and our result are closely related to his results. In the future, we will try to study -adic stochastic problems associated with our theorems. For example, -adic -Bernstein polynomials seem to be closely related to our results (see [6, 14, 20]).

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