Pseudo-factorials, elliptic functions, and continued fractions

TL;DR

This paper explores the properties of pseudo-factorials linked to elliptic functions, establishing their generating functions, orthogonal polynomials, and lattice sum expressions, revealing new mathematical structures.

Contribution

It introduces a new family of elliptic polynomials and characterizes pseudo-factorials through continued fractions and lattice sums, expanding understanding of their mathematical properties.

Findings

Established continued fraction expansion of pseudo-factorial generating function

Characterized a new family of elliptic orthogonal polynomials

Derived elementary congruences and lattice sum expressions

Abstract

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 ordinary generating function of pseudo-factorials, first discovered empirically, is established here. This article also provides a characterization of the associated orthogonal polynomials, which appear to form a new family of "elliptic polynomials", as well as various other properties of pseudo-factorials, including a hexagonal lattice sum expression and elementary congruences.