• A recurrence formula for leapi...

Result: A recurrence formula for leaping convergents of non-regular continued fractions

Title:
A recurrence formula for leaping convergents of non-regular continued fractions
Authors:
ELSNER, Carsten, KOMATSU, Takao
Source:
Linear algebra and its applications. 428(4):824-833
Publisher Information:
New York, NY: Elsevier Science, 2008.
Publication Year:
2008
Physical Description:
print, 8 ref
Original Material:
INIST-CNRS
Subject Terms:
Mathematics, Mathématiques, Sciences exactes et technologie, Exact sciences and technology, Sciences et techniques communes, Sciences and techniques of general use, Mathematiques, Mathematics, Combinatoire. Structures ordonnées, Combinatorics. Ordered structures, Combinatoire, Combinatorics, Algèbre, Algebra, Algèbre linéaire et multilinéaire, matrices, Linear and multilinear algebra, matrix theory, Arithmétique, Arithmetics, Aritmética, Combinatoire énumérative, Enumerative combinatorics, Fraction continue, Continued fractions, Généralisé, Generalized, Generalizado, Identité combinatoire, Combinatorial identity, Identidad combinatoria, Relation linéaire, Linear relation, Relación lineal, Relation récurrence, Recurrence relation, Relación recurrencia, Récurrence, Recurrence, Recurrencia, Régularité, Regularity, Regularidad, 05A19, 11A55, 11D04, 11J70, 15A06 Leaping convergents, Linear recurrence relations
Document Type:
Academic journal Article
File Description:
text
Language:
English
Author Affiliations:
Fachhochschule für die Wirtschaft, University of Applied Sciences, Freundallee 15, 30173 Hannover, Germany
Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan
ISSN:
0024-3795
Access URL:
http://pascal-francis.inist.fr/vibad/index.php?action=search&terms=20016497
Rights:
Copyright 2008 INIST-CNRS
CC BY 4.0
Sauf mention contraire ci-dessus, le contenu de cette notice bibliographique peut être utilisé dans le cadre d’une licence CC BY 4.0 Inist-CNRS / Unless otherwise stated above, the content of this bibliographic record may be used under a CC BY 4.0 licence by Inist-CNRS / A menos que se haya señalado antes, el contenido de este registro bibliográfico puede ser utilizado al amparo de una licencia CC BY 4.0 Inist-CNRS
Notes:
Mathematics
Accession Number:
edscal.20016497
Database:
PASCAL Archive

Further Information

Given a continued fraction [a0; a1, a2;... ], pn/qn = [a0: a1,.... an l is called the nth convergent for n = 0, 1, 2,... Leaping convergents are those of every rth convergent Prn+i/qrn+i (n = 0, 1, 2,...) for fixed integers r and i with r ≥ 2 and i = 0, 1,... r - 1. This leaping step r can be chosen as the length of period in the continued fraction. Elsner studied the leaping convergents p3n+ 1/q3n+1 for the continued fraction of e = [2; 1,2k, 1]∞k=1] and obtained some arithmetic properties. Komatsu studied those p3n/q3n for e1/s = [1; s(2k-1)-1, 1, 1]∞k=1 (s ≥ 2). He has also extended such results for some more general continued fractions. Such concepts have been generalized in the case of regular continued fractions. In this paper leaping convergents in the non-regular continued fractions are considered so that a more general three term relation is satisfied. Moreover, the leaping step r need not necessarily to equal the length of period. As one of applications a new recurrence formula for leaping convergents of Apery's continued fraction of ζ(3) is shown.