The free tangent law

Wiktor Ejsmont, Franz Lehner*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Nevanlinna-Herglotz functions play a fundamental role for the study of infinitely divisible distributions in free probability [11]. In the present paper we study the role of the tangent function, which is a fundamental Herglotz-Nevanlinna function [28,23,54], and related functions in free probability. To be specific, we show that the function [Formula presented] of Carlitz and Scoville [17, (1.6)] describes the limit distribution of sums of free commutators and anticommutators and thus the free cumulants are given by the Euler zigzag numbers.

Original languageEnglish
Article number102093
JournalAdvances in Applied Mathematics
Publication statusPublished - Oct 2020


  • Central limit theorem
  • Cotangent sums
  • Euler numbers
  • Free infinite divisibility
  • Tangent numbers
  • Zigzag numbers

ASJC Scopus subject areas

  • Applied Mathematics


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