Similar to the associated Legendre functions, the differential equation for the associated Bessel functions $B_{l,m}(x)$ is introduced so that its form remains invariant under the transformation $l\rightarrow -l-1$. A Rodrigues formula for the associated Bessel functions as squared integrable solutions in both regions $l<0$ and $l\geq 0$ is presented. The functions with the same $m$ but with different positive and negative values of $l$ are not independent of each other, while the functions with the same $l+m$ ($l-m$) but with different values of $l$ and $m$ are independent of each other. So, all the functions $B_{l,m}(x)$ may be taken into account as the union of the increasing (decreasing) infinite sequences with respect to $l$. It is shown that two new different types of exponential generating functions are attributed to the associated Bessel functions corresponding to these rearranged sequences.
Posgrado en Ciencias Físicas, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理科学研究生院）；Instituto de Ciencias Físicas Universidad Nacional Autónoma de México（墨西哥国立自治大学物理科学学院）
Applications in random matrix theory of a PIII$'$ $τ$-function sequence from Okamoto's Hamiltonian formulation
我们考虑由特征值的倒数之和组成的拉盖尔酉系综的奇异线性统计量。据观察，该统计量的指数生成函数可以写为一个Toeplitz行列式的条目中给出的特定$K$贝塞尔函数。早期的研究已经确定了相同的行列式，但用$I$贝塞尔函数代替了$K$贝塞尔函数，在非负整数拉盖尔参数的情况下，与拉盖尔酉系综的广义间隙概率的硬边标度极限有关。我们表明，Toeplitz行列式形成的任意线性组合，这两个贝塞尔函数发生作为一个$τ$ -函数序列冈本的哈密顿制定Painlevé III $'$，因此，对数导数的Toeplitz行列式满足相同的$σ$ -形式Painlevé III $'$微分方程，给出了一个事实，可以从早期的结果中观察到的解释。此外，一些见解之间的关系，这种特征的生成函数，其特征在$n \to \infty$限制，无论是与拉盖尔参数$α$固定，并与$α= n$（后一种情况是相关的应用程序的分布的维格纳时间延迟统计），给出。
