- Research Article
- Open Access

# Sums of Products of -Euler Polynomials and Numbers

- Young-Hee Kim
^{1}, - Kyung-Won Hwang
^{2}Email author and - Taekyun Kim
^{1}

**2009**:381324

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

© Young-Hee Kim et al. 2009

**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 .

## Keywords

- Positive Integer
- Generate Function
- Natural Number
- Complex Number
- Rational Number

## 1. Introduction

for all . Hence for any with (cf. [3–12]).

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

and the Euler numbers are defined as (cf. [1–20]).

where with . We note that and (cf. [3, 6–12, 15–17]).

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 [21–24], 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 .

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

Therefore we have the following theorem.

Theorem 2.1.

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.

By Theorem 2.2, we obtain

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

Theorem 2.3.

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

Corollary 2.4.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.