论文标题
第二类Ferrers功能的高斯超几何表示
Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind
论文作者
论文摘要
我们为第二类的Ferrers功能得出所有18个高斯超几何表示，每个都有不同的参数。它们是从第二种相关的Legendre函数的18个超几何表示，使用限制表示。对于与这些表示相对应的18个高几何参数，我们给出了复杂平面中相应收敛区域的几何描述。此外，我们考虑了第二种Ferrers功能的相应单和傅立叶扩展。在十种情况中的四种情况下，确定第二种的Ferrers功能需要评估分支机构上方和下方分别以$ [1，\ infty）$的分别评估。为了完成这些派生，我们使用众所周知的结果来推导其上方和下方的超几何函数的表达式。最后，我们对Richard Olbricht的1888年论文进行了详细评论，Richard Olbricht是第一个研究Legendre函数超几何表示的人。
We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the 18 hypergeometric arguments which correspond to these representations, we give geometrical descriptions of the corresponding convergence regions in the complex plane. In addition, we consider a corresponding single sum Fourier expansion for the Ferrers function of the second kind. In four of the eighteen cases, the determination of the Ferrers function of the second kind requires the evaluation of the hypergeometric function separately above and below the branch cut at $[1,\infty)$. In order to complete these derivations, we use well-known results to derive expressions for the hypergeometric function above and below its branch cut. Finally we give a detailed review of the 1888 paper by Richard Olbricht who was the first to study hypergeometric representations of Legendre functions.
