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 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.

Handedness and social anxiety: using Bryden's research as a catalyst to explore the influence of familial sinistrality and degree of handedness

Phil Bryden's work has impacted on many areas of laterality, including degree and measurement of hand preference, as well as influences of familial sinistrality (FS). For example, Bryden[(1977). Measuring handedness with questionnaires. Neuropsychologia, 15, 617–624] is a well-cited and influential paper that remains relevant to this day. Inspired by this we extended our analysis of the relationship between handedness and anxiety in a number of ways. We used familial handedness and strength of handedness to examine their potential influences on anxiety, and extended our research by exploring their relationship to social anxiety, using the Social Phobia Inventory (SPIN). Inconsistent left-handers (ILH) were found to be more socially anxious. In all categories of SPIN except avoidance, ILH were significantly more anxious than consistent right- and left-handers. There were FS differences between ILH with a first degree left-handed relative (FS+) compared to ILH with no first degree left-handed relative (FS−) on all categories of anxiety scores. Within FS+ participants, ILH had significantly higher anxiety scores, compared with consistent handers across all categories. This suggests that ILH's social anxiety may be influenced by a close left-handed relative. Inspired by examining Bryden's work for this special issue, we will continue to add both strength of preference and familial handedness to our work.