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arXiv 2103.02291math.CA
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L. L.的函数的渐近展开。卡拉舍娃

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

R B Paris

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10 pages, 2 figures
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摘要：

我们考虑$μ>0$，$0<σ<1$和$n=1, 2, \ldots\ $的整函数\[F_{n,σ}(x;μ)=\sum_{k=0}^\infty \frac{\sin\,(nγ_k)}{\sin γ_k}\,\frac{x^k}{k! Γ(μ-σk)},\quad γ_k=\frac{(k+1)π}{2n}\]的$x\to\pm\infty$的渐近展开.当$σ=α/(2n)$与$0<α<1$一起使用时，该功能最近由L. L. Karasheva [{ \it J. Math. Sciences}，{ \bf 250}（2020）753- 759]作为分数阶偏微分方程的解。通过将$F_{n,σ}(x;μ)$表示为Wright函数的有限和，我们利用超几何型积分函数的标准渐近性来确定其渐近展开。这被认为是关键取决于参数$σ$（并在较小程度上的整数$n$）。数值结果表明，所得到的不同的扩展的准确性。

英文摘要：

We consider the asymptotic expansion for $x\to\pm\infty$ of the entire function \[F_{n,σ}(x;μ)=\sum_{k=0}^\infty \frac{\sin\,(nγ_k)}{\sin γ_k}\,\frac{x^k}{k! Γ(μ-σk)},\quad γ_k=\frac{(k+1)π}{2n}\] for $μ>0$, $0<σ<1$ and $n=1, 2, \ldots\ $. When $σ=α/(2n)$, with $0<α<1$, 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 $F_{n,σ}(x;μ)$ as a finite sum of Wright functions, we employ the standard asymptotics of integral functions of hypergeometric type to determine its asymptotic expansion. This is 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.

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