Michael Obiero Oyengo

arXiv2018

Multiples of long period small element continued fractions to short period large elements continued fractions

We construct a class of quadratic irrationals having continued fractions of period $n\geq2$ with "small" partial quotients for which certain integer multiples have continued fractions of period $1$, $2$ or $4$ with "large" partial quotients. We then show that numbers in the period of the new continued fraction are simple functions of the numbers in the periods of the original continued fraction. We give generalizations of some of the continued fractions and show that polynomials arising from the generalizations are related to Chebyshev and Fibonacci polynomials. We then show that some of these polynomials have a hyperbolic root distribution.

math.NT↗

arXiv2018

Non-periodic continued fractions for quadratic irrationalities

A well known theorem of Lagrange states that the simple continued fraction of a real number $α$ is periodic if and only if $α$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two interlacing geometric series and show that in certain cases they converge to quadratic irrationalities. This phenomenon is connected with certain sequences of polynomials whose properties we examine further.

math.NT↗