The asymptotic expansion of a function introduced by L.L. Karasheva

Richard Paris*

*Corresponding author for this work

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    The asymptotic expansion for x→±∞ of the entire function Fn, σ(x; μ)=∑k=0∞sin(nγk)sinγkxkk!Γ(μ−σk), γk=(k+1)π2n for μ>0, 0σ1 and n=1, 2, … is considered. In the special case σ=α/(2n), with 0α1, this function was recently introduced by L.L. Karasheva (J. Math. Sciences, 250 (2020) 753–759) as a solution of a fractional-order partial differential equation. By expressing Fn, σ(x; μ) as a finite sum of Wright functions, we employ the standard asymptotics of integral functions of hypergeometric type to determine its asymptotic expansion. This was found to depend critically on the parameter σ (and to a lesser extent on the integer n). Numerical results are presented to illustrate the accuracy of the different expansions obtained.
    Original languageEnglish
    Article number1454
    Number of pages10
    Issue number12
    Early online date21 Jun 2021
    Publication statusPublished - 21 Jun 2021


    • Wright function
    • Asymptotic expansions
    • Stokes phenomenon


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