Pseudo-factorials, elliptic functions, and continued fractions
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.
Cite
@article{arxiv.0901.1379,
title = {Pseudo-factorials, elliptic functions, and continued fractions},
author = {Roland Bacher and Philippe Flajolet},
journal= {arXiv preprint arXiv:0901.1379},
year = {2009}
}
Comments
24 pages; with correction of typos and minor revision. To appear in The Ramanujan Journal