Deformed Convolution, Cumulant Transforms, and Semigroup Generators in Hurwitz Series Rings
Abstract
Let R be a commutative Q-algebra and let HR denote its Hurwitz series ring. We introduce a one-parameter convolution hconvt on HR for which the specialization t=−1 is the intrinsic Hurwitz product. Positive integral parameters act on finite-support truncations and reproduce finite free convolution in coefficient coordinates. A logarithmic transform yields additive and homogeneous cumulants, an explicit inversion formula, binomial, Hermite, Laguerre, and hypergeometric families, and coefficientwise analogues of the law of large numbers and the central limit theorem. At t=−1, addition of independent random variables becomes multiplication of their moment sequences in HR; this gives applications to classical cumulants, beta--gamma products, and self-decomposable laws. Weighted coefficient norms turn hconvt into a commutative Banach algebra product. Norm-continuous convolution semigroups then have generators conjugate to multiplication operators. Explicit formulas are obtained for the Hermite and Laguerre semigroups, the finite free heat generator, and the Lévy--Khintchine generator.
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BibTeX
@misc{ahmadi2026deformed,
title = {{Deformed Convolution, Cumulant Transforms, and Semigroup Generators in Hurwitz Series Rings}},
author = {Morteza Ahmadi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01555},
eprint = {2609.01555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01555},
}
RIS
TY - UNPB
TI - Deformed Convolution, Cumulant Transforms, and Semigroup Generators in Hurwitz Series Rings
AU - Morteza Ahmadi
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01555
UR - https://arxiv.org/abs/2609.01555
AB - Let R be a commutative Q-algebra and let HR denote its Hurwitz series ring. We introduce a one-parameter convolution hconvt on HR for which the specialization t=−1 is the intrinsic Hurwitz product. Positive integral parameters act on finite-support truncations and reproduce finite free convolution in coefficient coordinates. A logarithmic transform yields additive and homogeneous cumulants, an explicit inversion formula, binomial, Hermite, Laguerre, and hypergeometric families, and coefficientwise analogues of the law of large numbers and the central limit theorem. At t=−1, addition of independent random variables becomes multiplication of their moment sequences in HR; this gives applications to classical cumulants, beta--gamma products, and self-decomposable laws. Weighted coefficient norms turn hconvt into a commutative Banach algebra product. Norm-continuous convolution semigroups then have generators conjugate to multiplication operators. Explicit formulas are obtained for the Hermite and Laguerre semigroups, the finite free heat generator, and the Lévy--Khintchine generator.
ER -