Series and Products in the Development of Mathematics最新文献

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The Hypergeometric Series 超几何级数
Series and Products in the Development of Mathematics Pub Date : 1900-01-01 DOI: 10.1017/9781108709453.024
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引用次数: 0
Elliptic Functions: Nineteenth Century 椭圆函数:19世纪
Series and Products in the Development of Mathematics Pub Date : 1900-01-01 DOI: 10.1017/9781108671620.011
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引用次数: 0
Sums of Powers of Integers 整数幂的和
Series and Products in the Development of Mathematics Pub Date : 1900-01-01 DOI: 10.1017/9781108709453.003
F. T. Howard
{"title":"Sums of Powers of Integers","authors":"F. T. Howard","doi":"10.1017/9781108709453.003","DOIUrl":"https://doi.org/10.1017/9781108709453.003","url":null,"abstract":"and so on. Many different writers have worked on the problem of finding simple formulas for Skn, and many different methods have been used; see [3], [5], [6], [9], [10] for just a small sampling of recent articles. The formulas for Tk n are certainly less well known. In the present paper we use generating functions to find new recurrences for Skf7(a, d) and Tk n(a,d). We also show how Skn(a,d) and Tk n(a,d) can be determined from Sk_ln(a,d) and Tk_lr2(a, d), respectively, and we show how Skn(a, d) and Tkn(ay d) can be expressed in terms of Bernoulli numbers. One of the main results is a new \"lacunary\" recurrence formula for Skn with gaps of 6 (Theorem 3.1); that is, we can use the formula to find Smn for m = 0,1,..., 5; then, using only Smn, we can find S6+mt„; then, using only Smn and S6+nhn, we can find Si2+m3„, and so on. There is a similar recurrence for Tk^n. There are several motivations for this paper: (1) A recent article by Wiener [10] dealt with equation (1.1) and generalized some well-known properties of Skt„. We show how the formulas of [10] can be derived very quickly and how they can be extended. (2) In a recent article by Howard [4], formulas were found which connected Bernoulli numbers to Tkt„. Evidently, the properties of Tk^n are not well known, so the results of [4] are a stimulus to study Tk and Tkt„(a, d) in some detail. (3) The new lacunary recurrences mentioned above are useful and easy to use, and (in the writer's opinion) they are of considerable interest. In Section 3 we illustrate the formulas by computing S6n, Sl2}„, and^g^; in Section 6 we compute TAn,Tl0n, mdT16n. (4) Perhaps the main purpose of the paper is to show how generating functions provide a simple, unified approach to the study of sums of powers of integers. The many, and often repetitious","PeriodicalId":267892,"journal":{"name":"Series and Products in the Development of Mathematics","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132992489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 122
The Calculus of Newton and Leibniz 牛顿和莱布尼茨的微积分
Series and Products in the Development of Mathematics Pub Date : 1900-01-01 DOI: 10.1017/9781108709453.008
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引用次数: 0
Power Series in Fifteenth-Century Kerala 15世纪喀拉拉邦的幂级数
Series and Products in the Development of Mathematics Pub Date : 1900-01-01 DOI: 10.1017/9781108709453.002
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引用次数: 0
Infinite Product of Wallis Wallis的无穷积
Series and Products in the Development of Mathematics Pub Date : 1900-01-01 DOI: 10.1017/9781108709453.004
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引用次数: 0
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