On the asymptotics of wright functions of the second kind

Richard B. Paris, Armando Consiglio, Francesco Mainardi*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    2 Citations (Scopus)
    23 Downloads (Pure)


    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)=◀∑▶,Mσ(x)=∞∑n=0(−x)n/n!Γ(−nσ+1−σ)  (0<σ<1)


    for x → ± ∞ are presented. The situation corresponding to the limit σ → 1 is considered, where Mσ(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


    • Wright function
    • Auxiliary Wright function
    • Asymptotic expansions
    • Exponentially small expansions


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