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

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.