The Free Tangent Law
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866929364616085504 |
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| author | Ejsmont, Wiktor Lehner, Franz |
| author_facet | Ejsmont, Wiktor Lehner, Franz |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2004_02679 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The Free Tangent Law Ejsmont, Wiktor Lehner, Franz Operator Algebras Combinatorics Probability Primary: 46L54, Secondary: 11B68, 60F05 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. |
| title | The Free Tangent Law |
| topic | Operator Algebras Combinatorics Probability Primary: 46L54, Secondary: 11B68, 60F05 |
| url | https://arxiv.org/abs/2004.02679 |