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About: This article is published in American Mathematical Monthly.The article was published on 1997-04-01. It has received 3 citations till now.
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TL;DR: In this article, the generalized reciprocal logarithm numbers (GRLN) were derived for the Riemann zeta and gamma functions and various properties of the generalized GRLN were presented.
Abstract: Recently, a novel method based on the coding of partitions was used to determine a power series expansion for the reciprocal of the logarithmic function, viz. z/ln (1+z). Here we explain how this method can be adapted to obtain power series expansions for other intractable functions. First, the method is adapted to evaluate the Bernoulli numbers and polynomials. As a result, new integral representations and properties are determined for the former. Then via another adaptation of the method we derive a power series expansion for the function zs/ln s(1+z), whose polynomial coefficients Ak(s) are referred to as the generalized reciprocal logarithm numbers because they reduce to the reciprocal logarithm numbers when s=1. In addition to presenting a general formula for their evaluation, this paper presents various properties of the generalized reciprocal logarithm numbers including general formulas for specific values of s, a recursion relation and a finite sum identity. Other representations in terms of special polynomials are also derived for the Ak(s), which yield general formulas for the highest order coefficients. The paper concludes by deriving new results involving infinite series of the Ak(s) for the Riemann zeta and gamma functions and other mathematical quantities.
21 citations
TL;DR: The asymptotic statistical properties of a class of generalized discrepancies introduced by Cui and Freeden to test equidistribution on the sphere show that they have highly desirable properties and encompass several statistics already proposed in the literature.
Abstract: . In this paper, we derive the asymptotic statistical properties of a class of generalized discrepancies introduced by Cui and Freeden (SIAM J. Sci. Comput., 1997) to test equidistribution on the sphere. We show that they have highly desirable properties and encompass several statistics already proposed in the literature. In particular, it turns out that the limiting distribution is an (infinite) weighted sum of chi-squared random variables. Issues concerning the approximation of this distribution are considered in detail and explicit bounds for the approximation error are given. The statistics are then applied to assess the equidistribution of Hammersley low discrepancy sequences on the sphere and the uniformity of a dataset concerning magnetic orientations.
12 citations
05 Jan 1999
TL;DR: This paper describes mathematics writing, with particular emphasis on features of interest with respect to DDs, and discusses the strengths of DDs and some of the problems that need to be overcome before DDs can live up to claims made for them.
Abstract: The genre of mathematics writing has several distinctive features that point to some of the weaknesses of current digital documents (DDs). Some of these weaknesses are surprising. While it might be expected that the importance of formatting and special symbols in mathematics writing would pose challenges for DDs, the linked, chunked style of mathematics writing, with its theorems, lemmas, corollaries and remarks explicitly referring to each other, resembles standard hypertext so closely that one would expect that mathematics writing would take well to online hypertext form. It does not. This failure points to deficiencies in our understanding of the true strengths and weaknesses of DDs. This paper describes mathematics writing, with particular emphasis on features of interest with respect to DDs. The difficulties in producing effective mathematics DDs are examined and used as a basis for talking about general challenges for DDs. The paper then discusses the strengths of DDs and some of the problems that need to be overcome before DDs can live up to claims made for them. It also examines some of the misguided claims, such as superior support for nonlinearity, that are commonly made for DDs, explains why these claims are unwarranted and speculates on why the claims have been made anyway. Suggestions are given as to what the true benefits of digitization are, including performing computations on text, flexible control of time and better support for hiding information. The paper concludes with a list of questions whose answers are critical to understanding the capabilities, and therefore the future, of DDs.
4 citations