论文标题
Humbert Hyper-Bessel功能的渐近扩展
The asymptotic expansion of the Humbert hyper-Bessel function
论文作者
论文摘要
我们考虑使用$ {} _ 0f_2 $超测定功能表示的Humbert Hyper-Bessel函数的渐近扩展，\ [J_ {m，n}（x）= \ frac {（x/3）^{m+n}}}} {m！ n！} \，{} _ 0f_2（ - \！\！\！ - ; m+1，n+1; - （x/3）^3）\]作为$ x \ to+\ infty $，其中$ m $，$ n $不一定是非阴性整数。特别注意确定指数小的贡献。使用的主要方法是作者描述的（J.Comput。Appl。Math。{\ BF 234}（2010）488-504）;还可以通过应用于包含贝塞尔函数的积分表示的鞍点方法来获得领先的估计。提出了数值结果，以证明所得化合物扩展的准确性。
We consider the asymptotic expansion of the Humbert hyper-Bessel function expressed in terms of a ${}_0F_2$ hypergeometric function by \[J_{m,n}(x)=\frac{(x/3)^{m+n}}{m! n!}\,{}_0F_2(-\!\!\!-;m+1, n+1; -(x/3)^3)\] as $x\to+\infty$, where $m$, $n$ are not necessarily non-negative integers. Particular attention is paid to the determination of the exponentially small contribution. The main approach utilised is that described by the author (J. Comput. Appl. Math. {\bf 234} (2010) 488-504); a leading-order estimate is also obtained by application of the saddle-point method applied to an integral representation containing a Bessel function. Numerical results are presented to demonstrate the accuracy of the resulting compound expansion.
