标题： 第一类和第二类雅可比函数的双重求和加法公式 显示英文标题

标题： Double summation addition theorems for Jacobi functions of the first and second kind

摘要： 本文我们回顾并推导了第一类和第二类雅可比函数的双求和加法定理，涉及双曲和三角两种形式。与这些加法定理相关联，我们对雅可比函数的对称、反对称以及半整数奇参数值与满足二次变换的高斯超几何函数之间的关系进行了全面分析，其中包括相关的勒让德函数、盖根鲍尔函数以及第一类和第二类费雷尔斯函数。此外，我们介绍了雅可比函数的奥尔弗规范化形式，这对参数为整数时推导展开公式特别有用。我们还介绍了第二类雅可比函数的加法定理在拉普拉斯-贝尔特拉米算子的基本解的分离本征函数展开中的一个应用，这些空间包括紧致和非紧致的秩一对称空间。 显示英文摘要

摘要： In this paper we review and derive hyperbolic and trigonometric double summation addition theorems for Jacobi functions of the first and second kind. In connection with these addition theorems, we perform a full analysis of the relation between symmetric, antisymmetric and odd-half-integer parameter values for the Jacobi functions with certain Gauss hypergeometric functions which satisfy a quadratic transformation, including associated Legendre, Gegenbauer and Ferrers functions of the first and second kind. We also introduce Olver normalizations of the Jacobi functions which are particularly useful in the derivation of expansion formulas when the parameters are integers. We introduce an application of the addition theorems for the Jacobi functions of the second kind to separated eigenfunction expansions of a fundamental solution of the Laplace-Beltrami operator on the compact and noncompact rank one symmetric spaces.