分数阶Legendre函数的变换与求值
Legendre Functions of Fractional Degree: Transformations and Evaluations
摘要：
分数阶的相关勒让德函数出现在楔形或环形几何中的边值问题的解中，以及应用数学中的其他地方。在经典的情况下，当次数是半个奇数时，它们可以用完全椭圆积分表示。在这项研究中，许多转换，减少了情况下，不同的程度从一个整数的三分之一，四分之一或六分之一的经典情况。这些转换，或身份，方便的象征性操作和评价勒让德和费勒斯功能。他们推广了椭圆积分的Ramanujan变换和Whipple公式，后者将第一和第二类Legendre函数联系起来。证明采用了由代数曲线指定的代数坐标变换。
英文摘要：
Associated Legendre functions of fractional degree appear in the solution of boundary value problems in wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer, they can be expressed using complete elliptic integrals. In this study, many transformations are derived, which reduce the case when the degree differs from an integer by one-third, one-fourth or one-sixth to the classical case. These transformations, or identities, facilitate the symbolic manipulation and evaluation of Legendre and Ferrers functions. They generalize both Ramanujan's transformations of elliptic integrals and Whipple's formula, which relates Legendre functions of the first and second kinds. The proofs employ algebraic coordinate transformations, specified by algebraic curves.
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