Generalised hyperbolic polynomials: combinatorial interpretation and real-rootedness
Abstract
For p,q natural numbers, define Δ⟨ p,q ⟩:=xp−1∂/(∂ y) + yq−1∂/(∂ x), and consider the sequence of polynomials biggl((Δ⟨ p,q ⟩)n · xbiggr)n≥ 0; we call these the generalised hyperbolic polynomials. In this paper, we provide a simple combinatorial interpretation for these polynomials. As special cases, we obtain combinatorial interpretations for the coefficients of the analogs of sinh and cosh corresponding to the curve xp− p yq/q=1; on setting p=q=3, we obtain Conrad and Flajolet's tree interpretation for the coefficients of the Dixonian elliptic functions (2006) as a special case. Finally, we show that for any p,q≥ 3, after suitably normalising followed by substituting y=1, we obtain a sequence of univariate polynomials which are real-rooted.
Related
- The real-rootedness of the toric g-contribution polynomials
- Real-rootedness of Kazhdan--Lusztig polynomials of sparse paving matroids
- Real-rootedness of the Z-polynomials of sparse paving matroids and strict interlacing for uniform matroids
- Real-rootedness and ultra log-concavity of rank-two matroid Ehrhart h*-polynomials
- Binomial expansions of Jacobi-Stirling numbers and real-rootedness of Jacobi-Stirling descent polynomials
BibTeX
@misc{deb2026generalised,
title = {{Generalised hyperbolic polynomials: combinatorial interpretation and real-rootedness}},
author = {Bishal Deb},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37931},
eprint = {2610.37931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37931},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generalised hyperbolic polynomials: combinatorial interpretation and real-rootedness
AU - Bishal Deb
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37931
UR - https://arxiv.org/abs/2610.37931
AB - For p,q natural numbers, define Δ^⟨ p,q ⟩:=x^p−1∂/(∂ y) + y^q−1∂/(∂ x), and consider the sequence of polynomials biggl((Δ^⟨ p,q ⟩)^n · xbiggr)_n≥ 0; we call these the generalised hyperbolic polynomials. In this paper, we provide a simple combinatorial interpretation for these polynomials. As special cases, we obtain combinatorial interpretations for the coefficients of the analogs of sinh and cosh corresponding to the curve x^p− p y^q/q=1; on setting p=q=3, we obtain Conrad and Flajolet's tree interpretation for the coefficients of the Dixonian elliptic functions (2006) as a special case. Finally, we show that for any p,q≥ 3, after suitably normalising followed by substituting y=1, we obtain a sequence of univariate polynomials which are real-rooted.
ER -