260 USEFUL FORMULAE, CONNECTING LEGENDRE'S COEFFICIENTS, WHICH 

 therefore the constant term in P n is 



1_ n\(-l) r n(n-l) ... (?+ 1) _ (- l) r n\ 

 2 ll 'n\ ' ~ 1.2. 3~... r ~2 n (r\Y 



( -I fn \_ = (--if ]_. 3 . 5 ... (n-1) 



(2 . 4 . 6 ... ny 2 . 4. 6 ... n 



or when p. = 0, the value of 6r is 



n 



1 . 2 . 3 ... n = (-iy{l .3.5... (n-1)} 2 



1 . 3~. 5 ... (2w-lj "~ 1 . 3 . 5 ... (2n-l) 



Next, let n m = 2r, so that 



n + m = 2 (?i r). 



Then the coefficient of /A n+1 " in (ft'-l)" is 



n(n-l) ... (n-r+l) 



1 . 2 . 3 ... r ( ' ' 



therefore when p. = 0, the value of 



d m P n= l_ d^ m ,,_,(n + m)l n(n-l) ...(n-r+l) 



dp m ' 2 n n\ <V l+m vr v " 2" n\ ' 1.2.3 ... r 



And G m = - x M 



1.3.5... (2w-l) c^ m 



= (1}^ 1 3 5 ... (n-m-1) . 1.3. 5 ... (n + w-l) 



1.3.5... (2n-l) 



when m = 0, this reduces to 



( _ ,! {1.3.5... (n-1)} 2 

 V ' 1. 3\ 5.,. (2-l) ' 



as before. 



