8 papers · 1 filter
Strong -log-convexity of the -Hoggatt polynomials
Shane Chern, Wenle Shi
In this paper, three (-)log-convexity or concavity properties related to the -Hoggatt numbers are investigated. First, we show that the -Hoggatt transformation preserves t…
On the occurrence of congruence multiplicities between Ramanujan's theta functions
Shane Chern, Nicolas Allen Smoot, Dazhao Tang
Recently, the first and third authors initiated a study of arithmetical relationships between Ramanujan's theta functions and . In this work, we prove eight families…
Hankel determinants of Catalan-like sequences
Shane Chern, Wenle Shi
In this paper, we compute the (shifted) Hankel determinants of Catalan-like sequences, which arise naturally from the weighted enumerations of nonintersecting Motzkin meanders. Amo…
Convolutive sequences, II: Parametrizations
Shane Chern, Dennis Eichhorn, Shishuo Fu +1
In recent work, the authors defined a sequence to be -convolutive exactly if \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end…
Tadpole Nahm sum as a Wronskian
Shane Chern, Chanh Tran, Tanay Wakhare
We show that the tadpole Nahm sum is essentially a specialization of a series studied by Bartlett and Warnaar. This overlooked connection was first discovered by the internal model…
Proof of Cigler's conjecture on -Hoggatt numbers
Shane Chern, Wenle Shi
We prove the nonnegativity and palindromicity of a family of polynomials arising from -Hoggatt numbers. The nonnegativity is derived from Stanley's -partition theory thro…
Finite Kleshchev bipartitions and -trinomial coefficients
Shane Chern
The Kleshchev multipartitions arise in the representation theory for the Ariki-Koike algebras. In previous work, Li, Stanton, Xue, Yee, and the author considered a refined enumerat…
On conjectural fermionic formulas for the Macdonald index in Argyres-Douglas theories
Shane Chern, Chanh Tran, Tanay Wakhare
We prove a fermionic-bosonic duality relation for the Macdonald index in Argyres-Douglas theories of type , thereby yielding a conjectural fermionic formula due to…