On the asymptotic of Wright functions of the second kind

Abstract

The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book {\it Fractional Calculus and Waves in Linear Viscoelasticity}, (2010)], $$ F_σ(x)=\sum_{n=0}^\infty \frac{(-x)^n}{n! \g(-nσ)}~,\quad M_σ(x)=\sum_{n=0}^\infty \frac{(-x)^n}{n! \g(-nσ+1-σ)}\quad(0<σ<1)$$ for $x\to\pm\infty$ are presented. The situation corresponding to the limit $σ\to1^-$ is considered, where $M_σ(x)$ approaches the Dirac delta function $δ(x-1)$. Numerical results are given to demonstrate the accuracy of the expansions derived in the paper, together with graphical illustrations that reveal the transition to a Dirac delta function as $σ\to 1^-$.