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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 [{}_1Psi_1(rho,k; rho,0;x)= sum_{n=0}^inftyfrac{Gamma(k+rho n)}{Gamma(rho n)},frac{x^n}{n!}qquad (|x|<infty)] when the parameter $rhoin (-1,0)cup (0,infty)$ 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 ${}_1Psi_1(rho,k;rho,0;x)$ 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 ${}_0Psi_1(-!!!-; rho,0;x)$ with $rhoin (-1,0)cup (0,infty)$, 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 ${}_0Psi_1(-!!!-; pmrho, 0;cdot)$ 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.
In this work we obtain an approximate solution of the strongly nonlinear second order differential equation $frac{d^{2}u}{dt^{2}}+omega ^{2}u+alpha u^{2}frac{d^{2}u}{dt^{2}}+alpha uleft( frac{du}{dt}right)^{2}+beta omega ^{2}u^{3}=0$, describing the
We prove two multivariate $q$-binomial identities conjectured by Bousseau, Brini and van Garrel [Stable maps to Looijenga pairs, arXiv:2011.08830] which give generating series for Gromov-Witten invariants of two specific log Calabi-Yau surfaces. The
We present some distinct asymptotic properties of solutions to Caputo fractional differential equations (FDEs). First, we show that the non-trivial solutions to a FDE can not converge to the fixed points faster than $t^{-alpha}$, where $alpha$ is the
In this paper, we obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolution
Recently, various extensions and variants of Bessel functions of several kinds have been presented. Among them, the $(p,q)$-confluent hypergeometric function $Phi_{p,q}$ has been introduced and investigated. Here, we aim to introduce an extended $(p,