The Asymptotic Expansion of a Function Introduced by L.L. Karasheva

The asymptotic expansion for x → ± ∞ of the entire function F n , σ ( x ; μ ) = ∑ k = 0 ∞ sin ( n γ k ) sin γ k x k k ! Γ ( μ − σ k ) , γ k = ( k + 1 ) π 2 n for μ > 0 , 0 < σ < 1 and n = 1 , 2 , … is considered. In the special case σ = α / ( 2 n ) , with 0 < α < 1 , 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 F n , 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.