Euler polynomials and alternating sums of powers of integers

IF 0.7 Q3 EDUCATION & EDUCATIONAL RESEARCH

International Journal of Mathematical Education in Science and Technology Pub Date : 2023-07-03 DOI:10.1080/0020739X.2022.2074904

J. Cereceda

引用次数: 0

Abstract

In this note, we consider the alternating sum of the mth powers of the first n positive integers . In 1989, Gessel and Viennot showed that, for even m = 2k, can be expressed as a polynomial of degree k in the triangular number without constant term. Here, we offer an alternative demonstration of this result that can be made suitable for first-year undergraduate students by using some basic properties of the Euler numbers and polynomials. We also give the corresponding closed formula for in the case of odd powers m = 2k + 1. In addition, we express both and as polynomials in n and give their coefficients in terms of the Bernoulli numbers. Finally, we give yet another representation for and involving the Stirling numbers of the second kind.

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欧拉多项式和整数的交替幂和

在这个笔记中，我们考虑前n个正整数的m次幂的交替和。1989年，Gessel和Viennot证明，对于偶数m = 2k，可以表示为没有常数项的三角形数中的k次多项式。在这里，我们利用欧拉数和多项式的一些基本性质，为这一结果提供了另一种适合大一本科生的论证。在奇次幂m = 2k + 1的情况下，我们也给出了相应的封闭公式。此外，我们将两者都表示为n的多项式，并用伯努利数表示它们的系数。最后，我们给出了第二类斯特林数的另一种表示。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal of Mathematical Education in Science and Technology EDUCATION & EDUCATIONAL RESEARCH-

CiteScore

3.30

自引率

11.10%

发文量

123

期刊介绍： Mathematics is pervading every study and technique in our modern world, bringing ever more sharply into focus the responsibilities laid upon those whose task it is to teach it. Most prominent among these is the difficulty of presenting an interdisciplinary approach so that one professional group may benefit from the experience of others. The International Journal of Mathematical Education in Science and Technology provides a medium by which a wide range of experience in mathematical education can be presented, assimilated and eventually adapted to everyday needs in schools, colleges, polytechnics, universities, industry and commerce. Contributions will be welcomed from lecturers, teachers and users of mathematics at all levels on the contents of syllabuses and methods of presentation.