标题： 第二类雅可比函数乘积的积分表示 显示英文标题

标题： Integral representation for a product of two Jacobi functions of the second kind

摘要： 通过从Durand的双重积分表示法开始，该表示法用于两个第二类Jacobi函数的乘积，我们推导出第二类两个Jacobi函数乘积的核形式积分表示。我们还推导出第二类两个Jacobi函数乘积的Bateman型求和公式。从这个积分表示中，我们推导出第一类Jacobi函数在双曲和三角上下文中的积分表示。从Jacobi函数的积分表示中，我们还推导出一些极限函数的乘积的积分表示，例如第一类和第二类的关联Legendre函数、Ferrers函数以及第一类和第二类的Gegenbauer函数。通过检查这些乘积之一在相关函数奇点附近的性质，我们也推导出单个函数的积分表示，包括第二类Jacobi函数的Laplace型积分表示。最后，我们使用第二类函数的乘积公式，推导出第一类和第二类Jacobi函数的平方和的Nicholson型积分关系，并在合流极限下，得到第一类和第二类Laguerre函数的相应关系，这些关系推广了关系$\expe^{ix}\expe^{-ix}=1$到这些函数。 显示英文摘要

摘要： By starting with Durand's double integral representation for a product of two Jacobi functions of the second kind, we derive an integral representation for a product of two Jacobi functions of the second kind in kernel form. We also derive a Bateman-type sum for a product of two Jacobi functions of the second kind. From this integral representation we derive integral representations for the Jacobi function of the first kind in both the hyperbolic and trigonometric contexts. From the integral representations for Jacobi functions, we also derive integral representations for products of limiting functions such as associated Legendre functions of the first and second kind, Ferrers functions and also Gegenbauer functions of the first and second kind. By examining the behavior of one of these products near singularities of the relevant functions, we also derive integral representations for single functions, including a Laplace-type integral representation for the Jacobi function of the second kind. Finally, we use the product formulas for the functions of the second kind to derive Nicholson-type integral relations for the sums of squares of Jacobi functions of the first and second kinds, and in a confluent limit, Laguerre functions of the first and second kinds, which generalize the relation $\expe^{ix}\expe^{-ix}=1$ to those functions.