On the asymptotics of wright functions of the second kind

Richard B. Paris, Armando Consiglio, Francesco Mainardi*

*Corresponding author for this work

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    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)], Fσ(x)=∑n=0∞(−x)nn!Γ(−nσ) , Mσ(x)=∑n=0∞(−x)nn!Γ(−nσ+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 − .
    Original languageEnglish
    Pages (from-to)54-72
    Number of pages19
    JournalFractional Calculus and Applied Analysis
    Issue number1
    Early online date29 Jan 2021
    Publication statusPublished - 23 Feb 2021

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