135 research outputs found
New Finite Rogers-Ramanujan Identities
We present two general finite extensions for each of the two Rogers-Ramanujan
identities. Of these one can be derived directly from Watson's transformation
formula by specialization or through Bailey's method, the second similar
formula can be proved either by using the first formula and the q-Gosper
algorithm, or through the so-called Bailey lattice.Comment: 19 pages. to appear in Ramanujan
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 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.Comment: 24 pages; with correction of typos and minor revision. To appear in
The Ramanujan Journa
Rogers-Ramanujan Computer Searches
We describe three computer searches (in PARI/GP, Maple, and Mathematica, respectively) which led to the discovery of a number of identities of Rogers–Ramanujan type and identities of false theta functions
A unifying combinatorial approach to refined little G\"ollnitz and Capparelli's companion identities
Berkovich-Uncu have recently proved a companion of the well-known
Capparelli's identities as well as refinements of Savage-Sills' new little
G\"ollnitz identities. Noticing the connection between their results and
Boulet's earlier four-parameter partition generating functions, we discover a
new class of partitions, called -strict partitions, to generalize their
results. By applying both horizontal and vertical dissections of Ferrers'
diagrams with appropriate labellings, we provide a unified combinatorial
treatment of their results and shed more lights on the intriguing conditions of
their companion to Capparelli's identities.Comment: This is the second revision submitted to JCTA in June, comments are
welcom
Characters of graded parafermion conformal field theory
The graded parafermion conformal field theory at level k is a close cousin of
the much-studied Z_k parafermion model. Three character formulas for the graded
parafermion theory are presented, one bosonic, one fermionic (both previously
known) and one of spinon type (which is new). The main result of this paper is
a proof of the equivalence of these three forms using q-series methods combined
with the combinatorics of lattice paths. The pivotal step in our approach is
the observation that the graded parafermion theory -- which is equivalent to
the coset osp(1,2)_k/ u(1) -- can be factored as (osp(1,2)_k/ su(2)_k) x
(su(2)_k/ u(1)), with the two cosets on the right equivalent to the minimal
model M(k+2,2k+3) and the Z_k parafermion model, respectively. This
factorisation allows for a new combinatorial description of the graded
parafermion characters in terms of the one-dimensional configuration sums of
the (k+1)-state Andrews--Baxter--Forrester model.Comment: 36 page
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