Properties of Legendre functions - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 519

Those for which both degree and azimuthal number are variable are

1 ( n

1) P n − 1 ,

1

sinθ P m + 1 1) P n + 1 − = − m )( n − m + ( n + m )( n + m +

n

2 n +

(B.15)

1 P m + 1

1 ,

1

sinθ P n = − P m + 1 (B.16)

n − 1 n +

2 n +

P n − 1 = cosθ P n + 1) sinθ P m − n , ( n − m + (B.17)

sinθ P m + 1 1) P m − n . 2 mP n − 1 =− + ( n − m )( n − m + (B.18)

n

B.2 Evaluation of Legendre functions

We examine the evaluation of Legendre functions of the first kind only. For this
task (Press et al. , 1992, p. 247), we start with the closed expression (B.10) for the
sectoral harmonic P m ( z ),

2 m m ! 1

z 2 m /2 1) m (2 m )! P m ( z ) = ( − − . (B.19)

The factor (2 m )!/2 m m ! can be rearranged to give

(2 m )!
2 m m ! =

(2 m − 1)(2 m − 3) ··· (1) · (2 m )(2 m − 2) ··· (2)

2 m m !

= (2 m − 1)(2 m − 3) ··· (1)

= (2 m − 1)!!, (B.20)

a double factorial. It is thus the product of all positive odd integers less than 2 m .
By the recurrence relation (B.11),

( n − m + 1 ) P n + 1 = ( 2 n + 1 ) zP n − ( n + m ) P n − 1 .

(B.21)

This, then, relates the current Legendre function to the two Legendre functions of
the next lower degrees. For m = n ,

P m + 1 = 1) zP m ,

(2 m + (B.22)

since P n = 0for n < m . Thus, starting with P m as given by (B.19), we can generate

P m + 1 . Using (B.21), we can then generate P m + 2 , and so on, to the required P n .
The function subprogramme PMN(Z,M,N) calculates P n ( z ) for given values of
z and the integers m and n . The programme LEGENDRE.FOR is an interactive
programme that calls PMN(Z,M,N) to give the value of the Legendre function of
the first kind for azimuthal number m ,degree n and argument z .