Continued fractions and bounded analytic 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...

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

Elliptic Functions, Continued Fractions and Doubled Permutations

The question of the possible combinatorial significance of the integer coefficients appearing in the Taylor expansions of the Jacobian elliptic functions was first raised by Schiitzenberger. Indeed, these integers generalize the Euler numbers, i.e. the coefficients of tan(z), sec(z) whose relation to permutations has been known since the work of Andri: [l]. The first combinatorial interpretatio...