Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotic and structural properties of special cases of the Wright
  function arising in probability theory   This analysis paper presents previously unknown properties of some special
cases of the Wright function whose consideration is necessitated by our work on
probability theory and the theory of stochastic processes. Specifically, we
establish new asymptotic properties of the particular Wright function
\[{}_1\Psi_1(\rho,k; \rho,0;x)= \sum_{n=0}^\infty\frac{\Gamma(k+\rho
n)}{\Gamma(\rho n)}\,\frac{x^n}{n!}\qquad (|x|<\infty)\] when the parameter
$\rho\in (-1,0)\cup (0,\infty)$ and the argument $x$ is real. In the
probability theory applications, which are focused on studies of the
Poisson-Tweedie mixtures, the parameter $k$ is a non-negative integer. Several
representations involving well-known special functions are given for certain
particular values of $\rho$. The asymptotics of ${}_1\Psi_1(\rho,k;\rho,0;x)$
are obtained under numerous assumptions on the behavior of the arguments $k$
and $x$ when the parameter $\rho$ is both positive and negative.
  We also provide some integral representations and structural properties
involving the `reduced' Wright function ${}_0\Psi_1(-\!\!\!-; \rho,0;x)$ with
$\rho\in (-1,0)\cup (0,\infty)$, which might be useful for the derivation of
new properties of members of the power-variance family of distributions. Some
of these imply a reflection principle that connects the functions
${}_0\Psi_1(-\!\!\!-; \pm\rho, 0;\cdot)$ and certain Bessel functions.
  Several asymptotic relationships for both particular cases of this function
are also given. A few of these follow under additional constraints from
probability theory results which, although previously available, were unknown
to analysts.

22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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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: q-fin
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
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The categories of this paper are: eess