Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some properties of the free stable distributions   We investigate certain analytical properties of the free $\alpha-$stable
densities on the line. We prove that they are all classically infinitely
divisible when $\alpha\le 1$, and that they belong to the extended Thorin class
when $\alpha \leq 3/4.$ The L\'evy measure is explicitly computed for $\alpha
=1,$ showing that the free 1-stable random variables are not Thorin except in
the drifted Cauchy case. In the symmetric case we show that the free stable
densities are not infinitely divisible when $\alpha > 1.$ In the one-sided case
we prove, refining unimodality, that the densities are whale-shaped that is
their successive derivatives vanish exactly once. Finally, we derive a
collection of results connected to the fine structure of the one-sided free
stable densities, including a detailed analysis of the Kanter random variable,
complete asymptotic expansions at zero, a new identity for the Beta-Gamma
algebra, and several intrinsic properties of whale-shaped densities.

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The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg