Let
*x* = cos(), and
.
The associated Legendre functions
*P*_{n}^{m} = *P*_{n}^{m}(*x*) are listed below, for *n* = 0,8 and
*m* = 0,*n*,
in Table 3 (*n* = 0,5) and Table 4 (*n* = 6,8).
The associated Legendre functions with *m* = -*M* < 0 are simply given
in terms of the equivalent functions with *M* > 0 according to

(30) |

Table 3: List of associated Legendre polynomials from n = 0 to n = 5. |
||
---|---|---|

n | m | Function |

0 | 0 | 1 |

1 | 0 | x |

1 | 1 | s |

2 | 0 | |

2 | 1 | 3 x s |

2 | 2 | 3 (1-x^{2}) |

3 | 0 | |

3 | 1 | |

3 | 2 | 15 x (1-x^{2}) |

3 | 3 | 15 s^{3} |

4 | 0 | |

4 | 1 | |

4 | 2 | |

4 | 3 | 105xs^{3} |

4 | 4 | 105 s^{4} |

5 | 0 | |

5 | 1 | |

5 | 2 | |

5 | 3 | |

5 | 4 | 945xs^{4} |

5 | 5 | 945s^{5} |

Table 4:
List of associated Legendre polynomials from n = 6 to n =
8. |
||
---|---|---|

n | m | Function |

6 | 0 | |

6 | 1 | |

6 | 2 | |

6 | 3 | |

6 | 4 | |

6 | 5 | 10395xs^{5} |

6 | 6 | 10395s^{6} |

7 | 0 | |

7 | 1 | |

7 | 2 | |

7 | 3 | |

7 | 4 | |

7 | 5 | |

7 | 6 | 135,135xs^{6} |

7 | 7 | 135,135s^{7} |

8 | 0 | |

8 | 1 | |

8 | 2 | |

8 | 3 | |

8 | 4 | |

8 | 5 | |

8 | 6 | |

8 | 7 | 2,027,025xs^{7} |

8 | 8 | 2,027,025s^{8} |