Evaluation of toroidal harmonics

Abstract
Three algorithms to evaluate toroidal harmonics, i.e., Legendre functions of integral order and half-odd degree of the first and second kinds for real arguments larger than one, are presented. The first algorithm (DTORH1) allows the evaluation of the set {P_(n-1/2)
^m (x), Q_(n-1/2)
^m (x)} for fixed (integer and positive) values of m and n = 0, 1, ..., N. The algorithms DTORH2 and DTORH3 extend the method used in DTORH1 to obtain th...

Title of program: DTORH1, DTORH2, DTORH3
Catalogue Id: ADKV_v1_0

Nature of problem
We include three codes to evaluate toroidal harmonics: <ul type="circle"><li>DTORH1: This code evaluates toroidal harmonics (TH) of the first and second kinds for a given order m, from the lowest (positive) degreee (n=0) to a maximum degree n = N in the same run. <li>DTORH2: This code evaluates toroidal harmonics of the first and second kinds Pm(n-1/2)(x) and Qm(n-1/2)(x) for orders m = 0,...,M and degrees n = 0,...,N. In this code, for each given order m, TH up to the maximum degree reached by ...

Versions of this program held in the CPC repository in Mendeley Data
ADKV_v2_0; DTORH3 v 2.0; 10.1016/S0010-4655(01)00188-6
ADKV_v1_0; DTORH1, DTORH2, DTORH3; 10.1016/S0010-4655(99)00428-2

This program has been imported from the CPC Program Library held at Queen's University Belfast (1969-2019)

摘要
提出了三种用于计算环谐函数（toroidal harmonics）的算法，即针对实宗量大于1的整数阶、半奇数次第一类与第二类勒让德函数（Legendre functions）。第一种算法（DTORH1）可针对固定的（正整数）m与n=0,1,…,N，计算集合{P_{n-1/2}^m(x), Q_{n-1/2}^m(x)}。DTORH2与DTORH3则扩展了DTORH1所采用的方法，以获取……

程序名称：DTORH1、DTORH2、DTORH3
目录编号：ADKV_v1_0

问题属性
本文提供三款用于计算环谐函数的代码：
• DTORH1：该代码可在单次运行中，针对给定阶数m，计算从最低（正）次（n=0）至最高次n=N的第一类与第二类环谐函数。
• DTORH2：该代码可针对阶数m=0,…,M与次n=0,…,N，计算第一类与第二类环谐函数P_{n-1/2}^m(x)和Q_{n-1/2}^m(x)。在该代码中，针对每个给定阶数m，环谐函数的最高次可达……

本程序在孟德莱数据（Mendeley Data）的CPC程序库中留存的版本如下：
ADKV_v2_0；DTORH3 v 2.0；10.1016/S0010-4655(01)00188-6
ADKV_v1_0；DTORH1、DTORH2、DTORH3；10.1016/S0010-4655(99)00428-2

本程序源自贝尔法斯特女王大学所维护的CPC程序库（1969-2019）。