Open Access

Sums of Products of -Euler Polynomials and Numbers

Journal of Inequalities and Applications20092009:381324

https://doi.org/10.1155/2009/381324

Received: 4 March 2009

Accepted: 25 April 2009

Published: 7 June 2009

Abstract

We derive formulae for the sums of products of the -Euler polynomials and numbers using the multivariate fermionic -adic -Volkenborn integral on .

1. Introduction

The purpose of this paper is to derive formulae for the sums of products of the -Euler polynomials and numbers, since many identities can be obtained from our sums of products of the -Euler polynomials and numbers. In [1], Simsek evaluated the complete sums for the Euler numbers and polynomials and obtained some identities related to Euler numbers and polynomials from his complete sums, and Jang et al. [2] also considered the sums of products of Euler numbers. Kim [3] derived the sums of products of the -Euler numbers using the fermionic -adic -Volkenborn integral. In this paper, we will evaluate the complete sum of the -Euler polynomials and numbers using the fermionic -adic -Volkenborn integral on . Assume that is a fixed odd prime. Throughout this paper, the symbols and denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of , respectively. Let be the set of natural numbers. Let be the normalized exponential valuation of with When one talks about -extension, is variously considered as an indeterminate, which is a complex number or a -adic number . If one normally assumes If then one assumes We use the notations
(1.1)

for all . Hence for any with (cf. [312]).

For a fixed odd positive integer with , let
(1.2)
where lies in . The distribution on is defined by
(1.3)

We say that is a uniformly differentiable function at a point and denote this property by , if the difference quotients have a limit as .

For , the -deformed bosonic -adic integral is defined as
(1.4)
(see [12]). The fermionic -adic -measures on are defined as
(1.5)
and the -deformed fermonic -adic integral is defined by
(1.6)
(see [6]), for . The fermionic -adic integral on is defined as
(1.7)
It follows that where . If we take , then the classical Euler polynomials are defined by the generating function,
(1.8)

and the Euler numbers are defined as (cf. [120]).

It is known that the -Euler numbers are defined as
(1.9)
and the -Euler polynomials are defined as
(1.10)

where with . We note that and (cf. [3, 612, 1517]).

Let . We consider the -Euler numbers of order defined by
(1.11)
and the -Euler polynomials of order defined by
(1.12)
(see [57]). In Section 2, we evaluate the following multivariate fermionic -adic -integral :
(1.13)

for any elements , and distinct odd positive integers . We have the formulae for the complete sum of the products of -Euler polynomials related to the higher order -Euler polynomials using the fermionic -adic -Volkenborn integral on . We also obtain the formulae for the -Euler numbers.

In [2124], Khrennikov introduced other theories of -adic distributions which were recently generated in -adic mathematical physics, both bosonic and fermionic: Khrennikov tried to build a -adic picture of reality based on the field of -adic numbers and corresponding analysis (a particular case of so-called non-Archimedean analysis). He showed that many problems of the description of reality with the aid of real numbers are induced by unlimited application of the Archimedean axiom. This axiom means that the physical observation can be measured with an infinite exactness. The results connected with an infinite exactness of measurements appear all the time in the formalisms of quantum mechanics and quantum field theories, which have the real continuum as one of their foundations. In particular, the author explains that the famous EPR paradox is nothing other than a result of using ideal real elements corresponding to an infinite exactness of measurement of the position and the momentum of a quantum particle. From the author's point of view, the EPR paradox is only a new form of Zeno's ancient paradox of Achilles and the tortoise. Both of these paradoxes are connected with the notion of an infinitely deep and infinitely divisible real continuum (see [21, 22]). In [23, 24], Khrennikov outlines both the -adic frequency model and a measure-theoretic approach. The latter is understood in the sense of non-Archimedean integration theory where measures have only additive property, not -additive property, and satisfy a condition of the boundedness. Analogues of the laws of large numbers including the central limit theorem are given. They studied a possible statistical interpretation of group-valued probabilities as well as nontraditional probabilistic models in physics and the cognitive sciences.

2. Sums of Products of -Euler Polynomials and Numbers

Let , and let be distinct odd positive integers. Let be the least common multiple of .

Now we evaluate the multivariate fermionic -adic -integral
(2.1)

By the definition of the multivariate -adic -integral, we have

(2.2)
where . The second equality in (2.2) is satisfied by the equation
(2.3)
We easily see that
(2.4)
From (2.2), (2.4), and the definition of the -Euler polynomials, we derive the following equations:
(2.5)

Therefore we have the following theorem.

Theorem 2.1.

Let , and let . Let be distinct odd positive integers, and let be the least common multiple of . Then we have
(2.6)

By using the multinomial theorem, we can obtain the following Theorem 2.2. Theorem 2.2 is important to derive the main results of our paper.

Theorem 2.2.

Let and . Let be distinct odd positive integers. Then we have
(2.7)

By Theorem 2.2, we obtain

(2.8)

Hence we have the complete sum for q-Euler polynomials as follows.

Theorem 2.3.

Let and let . Let be distinct odd positive integers and be the least common multiple of . Then we have
(2.9)

When in Theorem 2.3, we obtain the following formula involving the -Euler numbers.

Corollary 2.4.

Let and . Let be distinct odd positive integers, and let be the least common multiple of . Then we have
(2.10)

Authors’ Affiliations

(1)
Division of General Education-Mathematics, Kwangwoon University
(2)
Department of General education, Kookmin University

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Copyright

© Young-Hee Kim et al. 2009

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