Formal power series of logarithmic type
type: Item
instance of: scholarly article
title: Formal power series of logarithmic type
zbMATH Open document ID: 0688.05001
DOI: 10.1016/0001-8708(89)90079-0
author: Daniel E. Loeb
author: Gian-Carlo Rota
published in: Advances in Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Synopsis of this 118 pages long paper (or better booklet) written by the authors has more than 14 pages, and shows the difficult role of the reviewer. A potential reader of this very lucidly written paper can perhaps obtain the first information about this work from these extracts from the authors' introduction: Since its beginning with Barrow and Newton, the calculus of finite differences has been viewed - whether admittedly or not - as a formal method of computation with special functions. Until nineteenth century, when the rigors of convergence were to be gradually imposed, mathematicians handling difference equations and series involving polynomials and logarithms, were not beset by doubts as to the correctness of their manipulations, and in fact, their results have seldom turned out to be incorrect, even by the standards of our day.... What is needed in order to justify formal manipulations - we can state nowadays with the confidence that comes after fifty years of local algebra - is an extension of the notion of formal power series that will include, besides powers of x, power series in other special functions of classical analysis, most notably exponentials and logarithms. It is the purpose of this work to carry out such an extension, one that includes the formal theory of infinite series in polynomials and logarithms. The difficulty of this program, and perhaps the reason why it has not been previously carried out, is the algebraic unwildiness of the functions \(x^ n(\log x)^ t\), where n is an integer and t a nonnegative integer. The coefficients in the Taylor expansions in power of a of the function \((x+a)(\log (x+a))^ t\) do not seem to follow rhyme or reason, and unless these coefficients are somehow made easy to handle, no simple operational calculus can be obtained. We resolve this difficulty by introducing another basis for the vector space spanned by the function \(x^ n(\log x)^ t\), a basis whose members we call the harmonic logarithms. We denote b…
reviewed by: Štefan Porubský
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05-02
zbMATH DE Number: 4127206
zbMATH Keywords: harmonic logarithms
zbMATH Keywords: Roman coefficients
MaRDI profile type: Publication
full work available at URL: 0001-8708(89)90079-0
OpenAlex ID: W1990741252
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description: scientific article; zbMATH DE number 4127206