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

TL;DR

This paper investigates the connection between Fibonacci ratios, 2-periodic continued fractions, and moment sequences, establishing conditions under which these sequences correspond to measures, including the Fibonacci ratios as moments.

Contribution

It demonstrates that certain finite continued fractions linked to 2-periodic fractions are moment sequences of measures, providing criteria for positivity, and specifically proves Fibonacci ratios are such moments.

Findings

Fibonacci ratios form moment sequences of measures.

Necessary and sufficient conditions for measure positivity are established.

Sequences associated with 2-periodic continued fractions are characterized as 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 F_{n+1}/F_{n+2}, n\ge 0 of consecutive Fibonacci numbers is a moment sequence.