Generalized Charlier Recurrence Coefficients: Hankel Formulas, Laguerre--Freud Dynamics, and Painlevé V Connections
Abstract
We study the monic recurrence coefficients of the generalized Charlier polynomials associated with the discrete weight ρμ(k) = μk/(k!)2, k∈ℕ0, μ>0. The coefficients βn and γn in the recurrence relation Pn+1(x) = (x−βn)Pn(x)−γnPn−1(x) are described through complementary moment, determinant, discrete dynamical, and continuous integrable formulations. We give a consistent normalization of the Markov function and its Jacobi continued fraction, and express the recurrence coefficients in terms of Hankel determinants, using a bordered determinant for the diagonal coefficient βn. Starting from the original Laguerre--Freud equations, we derive the local recursive system used for computation and relate it to the Toda deformation. We also recall the known Painlevé~V representations associated with the two-parameter generalized Charlier family, in a normalization consistent with the present specialization. Moment/Hankelconstructions provide complementary moment-theoretic realizations of the recurrence coefficients, while the local Laguerre--Freud recursion provides an independent dynamical construction. Their comparison is used for numerical cross-validation. The numerical results are consistent with the leading large-degree behavior βn∼ n, γn∼μ.
Related
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BibTeX
@misc{hounkonnou2026generalized,
title = {{Generalized Charlier Recurrence Coefficients: Hankel Formulas, Laguerre--Freud Dynamics, and Painlevé V Connections}},
author = {Mahouton Norbert Hounkonnou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35895},
eprint = {2609.35895},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35895},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generalized Charlier Recurrence Coefficients: Hankel Formulas, Laguerre--Freud Dynamics, and Painlevé V Connections
AU - Mahouton Norbert Hounkonnou
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.35895
UR - https://arxiv.org/abs/2609.35895
AB - We study the monic recurrence coefficients of the generalized Charlier polynomials associated with the discrete weight ρ_μ(k) = μ^k/(k!)^2, k∈ℕ_0, μ>0. The coefficients β_n and γ_n in the recurrence relation P_n+1(x) = (x−β_n)P_n(x)−γ_nP_n−1(x) are described through complementary moment, determinant, discrete dynamical, and continuous integrable formulations. We give a consistent normalization of the Markov function and its Jacobi continued fraction, and express the recurrence coefficients in terms of Hankel determinants, using a bordered determinant for the diagonal coefficient β_n. Starting from the original Laguerre--Freud equations, we derive the local recursive system used for computation and relate it to the Toda deformation. We also recall the known Painlevé~V representations associated with the two-parameter generalized Charlier family, in a normalization consistent with the present specialization. Moment/Hankelconstructions provide complementary moment-theoretic realizations of the recurrence coefficients, while the local Laguerre--Freud recursion provides an independent dynamical construction. Their comparison is used for numerical cross-validation. The numerical results are consistent with the leading large-degree behavior β_n∼ n, γ_n∼μ.
ER -