Continued fractions and irrational functions

Continued Fractions and Modular Functions

It is widely recognized that the work of Ramanujan deeply influenced the direction of modern number theory. This influence resonates clearly in the “Ramanujan conjectures.” Here I will explore another part of his work whose position within number theory seems to be less well understood, even though it is more elementary, namely that related to continued fractions. I will concentrate on the spec...

Continued Fractions, Fibonacci Numbers, and Some Classes of Irrational Numbers

In this paper we define an equivalence relation on the set of positive irrational numbers less than 1. The relation is defined by means of continued fractions. Equivalence classes under this relation are determined by the places of some elements equal to 1 (called essential 1’s) in the continued fraction expansion of numbers. Analysis of suprema of all equivalence classes leads to a solution wh...

Path generating functions and continued fractions

This paper extends Flajolet’s (Discrete Math. 32 (1980) 125-161) combinatorial theory of continued fractions by obtaining the generating function for paths between horizontal lines, with arbitrary starting and ending points and weights on the steps. Consequences of the combinatorial arguments used to determine this result are combinatorial proofs for many classical identities involving continue...

Pseudo-factorials, Elliptic Functions, and Continued Fractions

This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a Dixonian and a Weierstraß function, which parametrize the Fermat cubic curve and are relative to a hexagonal lattice. A continued fraction expansion of the ordin...

Orthogonal Rational Functions and Continued Fractions