Research Article | Open Access

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 -Genocchi Polynomials of Higher-Order and -Stirling Numbers by the Fermionic -Adic Integral on

**Academic Editor:**H. Srivastava

#### Abstract

A systemic study of some families of -Genocchi numbers and families of polynomials of NΓΆrlund 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 [1β10]. 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 [1β27]). Note that are referred to as the th Genocchi numbers. Let us now introduce the Genocchi polynomials of NΓΆrlund type as follows: (see [7, 9]). In the special case , and are referred to as the Genocchi numbers of NΓΆrlund 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 NΓΆrlund type. The purpose of this paper is to present a systemic study of some families -Genocchi numbers and polynomials of NΓΆrlund type by using the multivariate fermionic -adic integral on .

#### 2. β-Extensions of Genocchi Numbers and Polynomials of NΓΆrlund 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 NΓΆrlund 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 NΓΆrlund 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 NΓΆrlund 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 NΓΆrlund 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 NΓΆrlund 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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#### Copyright

Copyright © 2010 Seog-Hoon Rim 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.