The Free Tangent Law

Abstract

Nevanlinna-Herglotz functions play a fundamental role for the study of infinitely divisible distributions in free probability. In the present paper we study the role of the tangent function, which is a fundamental Herglotz-Nevanlinna function and related functions in free probability. To be specific, we show that the function $$\frac{\tan z}{1-x\tan z}$$ of Carlitz and Scoville describes the limit distribution of sums of free commutators and anticommutators and thus the free cumulants are given by the Euler zigzag numbers.

split off from arXiv:2002.06051;14 pages, 4 figures;final version to appear in Adv Appl Math

References in corpus (6)

• The Trace Method for Cotangent Sums
• Sums of Commutators in Free Probability
• Higher-order tangent and secant numbers
• Supports, regularity, and $⊞$-infinite divisibility for measures of the form $({\mu }^{⊞p}{)}^{\uplus q}$
• Inevitable Dottie Number. Iterals of cosine and sine
• Remarks on the tangent function from an analytic and probability point of view

Cited by in corpus (3)

• The Trace Method for Cotangent Sums
• Sums of Commutators in Free Probability
• On the anti-commutator of two free random variables