On the asymptotics of wright functions of the second kind

Abstract

Abstract The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book Fractional Calculus and Waves in Linear Viscoelasticity (2010)], F σ ( x ) = ∑ n = 0 ∞ ( − x ) n n ! Γ ( − n σ )   ,   M σ ( x ) = ∑ n = 0 ∞ ( − x ) n n ! Γ ( − n σ + 1 − σ )   ( 0 < σ < 1 ) $$F_\sigma(x)=\sum\limits_{n=0}^\infty \frac{(-x)^n}{n! {\mathrm{\Gamma}}(-n\sigma)}~,\quad M_\sigma(x)=\sum\limits_{n=0}^\infty \frac{(-x)^n}{n! {\mathrm{\Gamma}}(-n\sigma+1-\sigma)}\quad(0 \lt \sigma \lt 1) $$ for x → ± ∞ are presented. The situation corresponding to the limit σ → 1− 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 → 1−.