Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSGenerating functionThe generating function for associated Legendre functions can be easily derivedby combining their definition (18.32) with the generating function for the Legendrepolynomials given in (18.15). We find that(2m)!(1 − x 2 ) m/2G(x, h) =2 m m!(1 − 2hx + h 2 ) = ∑∞ Pn+m(x)h m n . (18.40)m+1/2n=0◮Derive the expression (18.40) for the associated Legendre generating function.The generating function (18.15) for the Legendre polynomials reads∞∑P n h n =(1− 2xh + h 2 ) −1/2 .n=0Differentiating both sides of this result m times (assumimg m to be non-negative), mutliplyingthrough by (1 − x 2 ) m/2 and using the definition (18.32) of the associated Legendrefunctions, we obtain∞∑Pn m h n =(1− x 2 m/2dm)dx (1 − 2xh + m h2 ) −1/2 .n=0Performing the derivatives on the RHS gives∞∑Pn m h n = 1 · 3 · 5 ···(2m − 1)(1 − x2 ) m/2 h m.(1 − 2xh + h 2 ) m+1/2 n=0Dividing through by h m , re-indexing the summation on the LHS and noting that, quitegenerally,1 · 3 · 5 ···(2r − 1) = 1 · 2 · 3 ···2r2 · 4 · 6 ···2r = (2r)!2 r r! ,we obtain the final result (18.40). ◭Recurrence relationsAs one might expect, the associated Legendre functions satisfy certain recurrencerelations. Indeed, the presence of the two indices n and m means that a much widerrange of recurrence relations may be derived. Here we shall content ourselveswith quoting just four of the most useful relations:Pn m+1 2mx=(1 − x 2 ) P m 1/2 n +[m(m − 1) − n(n +1)]Pn m−1 , (18.41)(2n +1)xPn m =(n + m)Pn−1 m +(n − m +1)Pn+1, m (18.42)(2n + 1)(1 − x 2 ) 1/2 Pn m = Pn+1 m+1 − P n−1 m+1 , (18.43)2(1 − x 2 ) 1/2 (Pn m ) ′ = Pnm+1 − (n + m)(n − m +1)Pn m−1 . (18.44)We note that, by virtue of our adopted definition (18.32), these recurrence relationsare equally valid for negative and non-negative values of m. These relations may592