Umbral Hermite representations of canonical catastrophe integrals
Abstract
Higher-order multivariable Hermite-Kampé de Fériet polynomials provide a unified construction of umbral representations for canonical catastrophe integrals. Expanding the unfolding exponential in these polynomials and integrating against the leading catastrophe kernel gives convergent moment expansions on suitable decay contours. A universal formal umbral operator then expresses the resulting fields through explicit moment ground states. We develop this construction for the fold, cusp and swallowtail, and relate the Euclidean Pearcey field to quartic scalar partition functions. The distinction between formal and analytic umbral frameworks becomes essential when integration is performed before umbral evaluation. For Euclidean Pearcey, the integrated linearized kernel is a rational function whose action is not specified by the original formal rules. The Analytic Umbral Framework supplies a convergent Mellin pairing for this kernel and recovers the independently obtained bivariate Pearcey representation. This establishes, in this example, an exchange of integration and analytic evaluation and identifies a specific extension beyond the formal framework.
Related
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BibTeX
@misc{dattoli2026umbral,
title = {{Umbral Hermite representations of canonical catastrophe integrals}},
author = {Giuseppe Dattoli and Roberto Ricci},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03809},
eprint = {2610.03809},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03809},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Umbral Hermite representations of canonical catastrophe integrals
AU - Giuseppe Dattoli
AU - Roberto Ricci
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.03809
UR - https://arxiv.org/abs/2610.03809
AB - Higher-order multivariable Hermite-Kampé de Fériet polynomials provide a unified construction of umbral representations for canonical catastrophe integrals. Expanding the unfolding exponential in these polynomials and integrating against the leading catastrophe kernel gives convergent moment expansions on suitable decay contours. A universal formal umbral operator then expresses the resulting fields through explicit moment ground states. We develop this construction for the fold, cusp and swallowtail, and relate the Euclidean Pearcey field to quartic scalar partition functions. The distinction between formal and analytic umbral frameworks becomes essential when integration is performed before umbral evaluation. For Euclidean Pearcey, the integrated linearized kernel is a rational function whose action is not specified by the original formal rules. The Analytic Umbral Framework supplies a convergent Mellin pairing for this kernel and recovers the independently obtained bivariate Pearcey representation. This establishes, in this example, an exchange of integration and analytic evaluation and identifies a specific extension beyond the formal framework.
ER -