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This analysis paper presents previously unknown properties of some specialcases of the Wright function whose consideration is necessitated by our work onprobability theory and the theory of stochastic processes. Specifically, weestablish new asymptotic properties of the particular Wright function\[{}_1\Psi_1(\rho,k; \rho,0;x)= \sum_{n=0}^\infty\frac{\Gamma(k+\rhon)}{\Gamma(\rho n)}\,\frac{x^n}{n!}\qquad (|x|<\infty)\] when the parameter$\rho\in (-1,0)\cup (0,\infty)$ and the argument $x$ is real. In theprobability theory applications, which are focused on studies of thePoisson-Tweedie mixtures, the parameter $k$ is a non-negative integer. Severalrepresentations involving well-known special functions are given for certainparticular 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 propertiesinvolving 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 ofnew properties of members of the power-variance family of distributions. Someof 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 functionare also given. A few of these follow under additional constraints fromprobability theory results which, although previously available, were unknownto analysts.

Asymptotic and structural properties of special cases of the Wright function arising in probability theory