Telescoping continued fractions for the error term in Stirling’s formula

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory Pub Date : 2023-09-01 DOI:10.1016/j.jat.2023.105943

Gaurav Bhatnagar , Krishnan Rajkumar

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引用次数: 0

Abstract

In this paper, we introduce telescoping continued fractions to find lower bounds for the error term $r_{n}$ in Stirling’s approximation $n! = \sqrt{2 π} n^{n + 1 / 2} e^{- n} e^{r_{n}} .$ This improves lower bounds given earlier by Cesàro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction. The technique we introduce allows us to experimentally obtain upper and lower bounds for a sequence of convergents of a continued fraction in terms of a difference of two continued fractions.

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斯特林公式中误差项的伸缩连分式

在本文中，我们引入伸缩连分式来寻找Stirling近似n中误差项rn的下界=2πnn+1/2e−nern。这改进了Cesàro（1922）、Robbins（1955）、Nanjundiah（1959）、Maria（1965）和Popov（2017）早些时候给出的下限。该表达式是以连续分数的形式表示的，以及找到该连续分数的连续项的算法。我们引入的技术使我们能够通过实验获得连续分数的收敛序列的上界和下界，这些收敛序列是根据两个连续分数的差来确定的。

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

Journal of Approximation Theory 数学-数学

CiteScore

1.90

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.