Explicit continued fractions related to Fibonacci sequences

Abstract

In 1977, Davison discovered a remarkable class of continued fractions for real numbers representable as sums of reciprocals of powers of 2; these powers being taken from the sequence of the integral parts of multiples of the golden number, a root of the characteristic polynomial of the Fibonacci re-currence sequence. Soon after, Adams and Davison generalized this result by replacing 2 with any integer greater than 1. In 1986, Bowman using a different approach based partly on unique representation of positive integers by sums of integers belonging to the Fibonacci sequence, known as Zeckendorff expansion, generalized Davison's result even further by obtaining explicit continued fractions for numbers representable as quotients of series of the above form. In this thesis, we deal with the problem of generalizing Bowman's results by extending Zeckendorff expansion to expansion via the (h, k)[supscript th] Fi-bonacci sequence introduced by Daykin; the (2,2)-class is that of Fibonacci sequence. Explicit continued fractions are derived for real numbers which are quotients of series of terms whose powers are representable via (h, k)-class expansions.