Free Probability Theory Over the Scaled Hyperbolic Numbers

Authors

Daniel Alpay, Chapman UniversityFollow
Ilwoo Choo, St. Ambrose UniversityFollow

Document Type

Article

Publication Date

12-12-2025

Abstract

In this paper, we introduce a notion of free probability over the scaled hyperbolic numbers. Scaled hypercomplex numbers {D_t} _t∈R are constructed as sub-structures of scaled hypercomplex numbers {H_t} _t∈R under the scales (or, the moments) of the set R of real numbers.We show that if t < 0, then the classical free probability theory covers our free probability on {D_t} _{t< 0}; if t > 0, then our free probability on {D_t} _t>0 is represented by the free probability over the classical hyperbolic numbers D = D₁; and if t = 0, then the free probability on D₀ is actually over the dual numbers D = D₀. Since the usual free probability theory is over C, we here concentrate on establishing our free probability theory on D, or that on D. Our approaches are motivated by the Speicher’s combinatorial free probability. As applications, the D_t-free-probabilistic versions of semicircular elements and circular elements are considered.

Comments

This article was originally published in Advances in Applied Clifford Algebras, volume 36, in 2026. https://doi.org/10.1007/s00006-025-01427-1

Recommended Citation

Alpay, D., Cho, I. Free Probability Theory over the Scaled Hyperbolic Numbers. Adv. Appl. Clifford Algebras 36, 7 (2026). https://doi.org/10.1007/s00006-025-01427-1

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This work is licensed under a Creative Commons Attribution 4.0 License.