The asymptotic expansion of a function due to L.L. Karasheva

We consider the asymptotic expansion for $x\to\pm\infty$ of the entire function $$F_{n,\sigma}(x;\mu)=\sum_{k=0}^\infty \frac{\sin\,(n\gamma_k)}{\sin \gamma_k}\,\frac{x^k}{k! \Gamma(\mu-\sigma k)},\quad \gamma_k=\frac{(k+1)\pi}{2n}$$ for $\mu>0$, $0<\sigma<1$ and $n=1, 2, \ldots\$. When $\sigma=\alpha/(2n)$, with $0<\alpha<1$, this function was recently introduced by L.L. Karasheva [{\it J. Math. Sciences}, {\bf 250} (2020) 753--759] as a solution of a fractional-order partial differential equation. By expressing $F_{n,\sigma}(x;\mu)$ as a finite sum of Wright functions, we employ the standard asymptotics of integral functions of hypergeometric type to determine its asymptotic expansion. This is found to depend critically on the parameter $\sigma$ (and to a lesser extent on the integer $n$). Numerical results are presented to illustrate the accuracy of the different expansions obtained.