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

Richard B. Paris, Vladimir Vinogradov

    Research output: Contribution to journalArticlepeer-review

    10 Citations (Scopus)
    106 Downloads (Pure)


    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 1Ψ1(ρ, k; ρ, 0; x) = X∞ n=0 Γ(k + ρn) Γ(ρn) x n n! (|x| <∞) when the parameter ρ ∈ (−1, 0)∪(0, ∞) 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 ρ. The asymptotics of 1Ψ1(ρ, k; ρ, 0; x) are obtained under numerous assumptions on the behavior of the arguments k and x when the parameter ρ is both positive and negative. We also provide some integral representations and structural properties involving the ‘reduced’ Wright function 0Ψ1(−−; ρ, 0; x) with ρ ∈ (−1, 0) ∪ (0, ∞), 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 0Ψ1(−−;±ρ, 0; ·) 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.
    Original languageEnglish
    Pages (from-to)377-409
    Number of pages33
    JournalLithuanian Mathematical Journal
    Issue number3
    Publication statusPublished - 25 Jul 2016


    • Multiplication property
    • Wright function
    • Asymptotics
    • Exponentially small expansions
    • Algebraic expansions
    • Stokes lines
    • Reflection principle
    • Airy function
    • Bessel functions
    • Confluent hypergeometric function
    • Whittaker function


    Dive into the research topics of 'Asymptotic and structural properties of special cases of the Wright function arising in probability theory'. Together they form a unique fingerprint.

    Cite this