Fibonacci numbers, Euler’s 2-periodic continued fractions and moment sequences

Christian Berg; Antonio J. Durán

Fibonacci numbers, Euler’s 2-periodic continued fractions and moment sequences

Department of Mathematical Sciences

Abstract

We prove that certain sequences of finite continued fractions associated with a 2-periodic continued fraction with period a,b > 0 are moment sequences of discrete signed measures supported in the interval [-1,1], and we give necessary and sufficient conditions in order that these measures are positive. For a = b = 1 this proves that the sequence of ratios F_n+1/F _n+2, n ≥ 0, of consecutive Fibonacci numbers is a moment sequence.

Original language	English
Journal	Fibonacci Quarterly
Volume	49
Pages (from-to)	66-75
ISSN	0015-0517
Publication status	Published - Feb 2011

Cite this

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title = "Fibonacci numbers, Euler{\textquoteright}s 2-periodic continued fractions and moment sequences",

abstract = "We prove that certain sequences of finite continued fractions associated with a 2-periodic continued fraction with period a,b > 0 are moment sequences of discrete signed measures supported in the interval [-1,1], and we give necessary and sufficient conditions in order that these measures are positive. For a = b = 1 this proves that the sequence of ratios Fn+1/F n+2, n ≥ 0, of consecutive Fibonacci numbers is a moment sequence.",

author = "Christian Berg and Dur{\'a}n, {Antonio J.}",

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month = feb,

language = "English",

volume = "49",

pages = "66--75",

journal = "Fibonacci Quarterly",

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TY - JOUR

T1 - Fibonacci numbers, Euler’s 2-periodic continued fractions and moment sequences

AU - Berg, Christian

AU - Durán, Antonio J.

PY - 2011/2

Y1 - 2011/2

N2 - We prove that certain sequences of finite continued fractions associated with a 2-periodic continued fraction with period a,b > 0 are moment sequences of discrete signed measures supported in the interval [-1,1], and we give necessary and sufficient conditions in order that these measures are positive. For a = b = 1 this proves that the sequence of ratios Fn+1/F n+2, n ≥ 0, of consecutive Fibonacci numbers is a moment sequence.

AB - We prove that certain sequences of finite continued fractions associated with a 2-periodic continued fraction with period a,b > 0 are moment sequences of discrete signed measures supported in the interval [-1,1], and we give necessary and sufficient conditions in order that these measures are positive. For a = b = 1 this proves that the sequence of ratios Fn+1/F n+2, n ≥ 0, of consecutive Fibonacci numbers is a moment sequence.

M3 - Journal article

SN - 0015-0517

VL - 49

SP - 66

EP - 75

JO - Fibonacci Quarterly

JF - Fibonacci Quarterly

ER -

Fibonacci numbers, Euler’s 2-periodic continued fractions and moment sequences

Abstract

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