A simple algorithm for expanding a power series as a continued fraction

被引：3

作者：

Sokal, Alan D. ^{[1
,2
]}

机构：

[1] UCL, Dept Math, London WC1E 6BT, England

[2] NYU, Dept Phys, 726 Broadway, New York, NY 10003 USA

来源：

EXPOSITIONES MATHEMATICAE | 2023年 / 41卷 / 02期

基金：

英国工程与自然科学研究理事会;

关键词：

Formal power series; Continued fraction; Euler-Viscovatov algorithm; Gauss's continued fraction; Euler-Gauss recurrence method; Motzkin path; Dyck path; Stieltjes table; Rogers' addition formula; HYPERGEOMETRIC-SERIES; ELLIPTIC FUNCTIONS; PROOF;

D O I：

10.1016/j.exmath.2022.12.001

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980).& COPY; 2022 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

引用

页码：245 / 287

页数：43

共 50 条

[1] A simple algorithm for expanding a power series as a continued fraction
Sokal, Alan D.
arXiv, 2022,
[2] GENERALIZATION OF SIMPLE CONTINUED FRACTION ALGORITHM
VAUGHAN, TP
MATHEMATICS OF COMPUTATION, 1978, 32 (142) : 537 - 558
[3] NOTE ON THE EXPANSION OF A POWER SERIES INTO A CONTINUED FRACTION
WALL, HS
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 1945, 51 (02) : 97 - 105
[4] Continued fraction expansions for arbitrary power series
Scott, WT
Wall, HS
ANNALS OF MATHEMATICS, 1940, 41 : 328 - 349
[5] The convergence of an associated continued fraction with a power series
Hamburger, H
MATHEMATISCHE ANNALEN, 1920, 81 : 31 - 45
[6] A certain power series and the inhomogeneous continued fraction expansions
Komatsu, T
JOURNAL OF NUMBER THEORY, 1996, 59 (02) : 291 - 312
[7] On lacunary formal power series and their continued fraction expansion
France, MM
van der Poorten, AJ
Shallit, J
NUMBER THEORY IN PROGRESS, VOLS 1 AND 2: VOL 1: DIOPHANTINE PROBLEMS AND POLYNOMIALS; VOL 2: ELEMENTARY AND ANALYTIC NUMBER THEORY;, 1999, : 321 - 326
[8] Scaling in simple continued fraction
Yadav, Avinash Chand
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 2020, 53 (06)
[9] ON THE CONTINUED-FRACTION EXPANSION OF SOME FORMAL POWER-SERIES
ALLOUCHE, JP
COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE, 1988, 307 (12): : 631 - 633
[10] GENERALIZATION OF CONTINUED FRACTION ALGORITHM
GUTING, R
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 1976, 283 : 384 - 387

← 1 2 3 4 5 →