258 USEFUL FORMULAE, CONNECTING LEGENDRE'S COEFFICIENTS, WHICH 

 li). We have seen that 



P _ _ L ^10^1)! 



" 2". 1 . 2 . 3 ... n dp" 

 The coefficient of p." in this expression is 



2n (2n-l)(2n-2) ... (n+1) 2n(2n-l) ...3.2.1 _ !_. 3 . 5 ... (2n- 1) 

 ~2"Tl .2.3 ...1T~ 2"(1 . 2 . 3 ... w)* ' ~TT2~. 3 ... n~ 



rf"P 



Hence the coefficient of u," "' in -p is 



f/jU, 



1.3.5... (2-l) 1.3.5... (2n- 1) 



(n-l) ... (n-m+l)x- -V- -(. 



1 . 2 . 3 ... n 1 . 2 . 3 ... (n m) 



n 



Also the coefficient of /u." '" in the value of Gr is unity. 



20. 2" n ! P n j (M" 1 )" 



Differentiating m times with respect to p., then 



/ 



hence 2" ,* ! <X'= (1 -^ 



cti 



/f).,\! 



' 



_ r 



fl ?-( 1 - rt " v ~ 1)r ............... <23) - 







(n-m)(rt-m-l)(n-m-2)(n-m-3) 

 2.4.(2w-l)(2n-3) 



