论文标题
用于指数和贝塞尔函数的乘积之和的扩展。 ii
An expansion for the sum of a product of an exponential and a Bessel function. II
论文作者
论文摘要
我们检查了衰减指数的总和（取决于求和索引）和bessel函数的总和，以\ [\ sum_ {n = 1}^\ infty e^{ - an^p} \ an^p} \ frac {j_ n = 1}^\ frac { $ a \ to0 $，其中$j_ν（z）$是第一种真实订单$ν$，$ a $和$ p $的贝塞尔函数是正参数。通过Mellin Transform方法，我们获得了一个渐近扩展，该扩展可以在限制$ a \至0 $中评估该总和。当Bessel函数被修改的Bessel函数$I_ν（z）$替换时，总和得出了类似的结果。甚至$ p $的情况都引起了人们的关注，因为该扩展的性质呈指数级。我们证明，在$ p = 2 $的情况下，结果类似于上述总和的泊松 - 雅各比转换。
We examine the sum of a decaying exponential (depending non-linearly on the summation index) and a Bessel function in the form \[\sum_{n=1}^\infty e^{-an^p}\frac{J_ν(an^px)}{(an^px/2)^ν}\qquad (x>0),\] in the limit $a\to0$, where $J_ν(z)$ is the Bessel function of the first kind of real order $ν$ and $a$ and $p$ are positive parameters. By means of a Mellin transform approach we obtain an asymptotic expansion that enables the evaluation of this sum in the limit $a\to 0$. A similar result is derived for the sum when the Bessel function is replaced by the modified Bessel function $I_ν(z)$ when $x\in (0,1)$. The case of even $p$ is of interest since the expansion becomes exponentially small in character. We demonstrate that in the case $p=2$, a result analogous to the Poisson-Jacobi transformation exists for the above sum.
