252 USEFUL FORMULAE, CONNECTING LEGENDRE'S COEFFICIENTS, WHICH 



m 

 Multiply by (1 /r) 2 and we get 



+ (n + m) (n + m+l) (, .................. (11). 



If P n contain only even or only odd powers of p., and if \P n dfi 



vanishes when /t=l, then it also vanishes when /*=!, and \P n dp. will 

 contain only odd or even powers of /A respectively. 



First suppose n to be even and therefore P n to contain only even 

 powers of p, then \P n d^ will contain odd powers of /n together with a 



constant term, but since \P n d^ vanishes both when p.= l and when p,= 1 



this constant is zero, because the value of the remaining terms, if finite, 

 would change sign when p. is changed from 1 to 1 . 



Hence \P n d[j. contains only odd powers of /u, and is divisible by (1 /x, 2 ). 



Secondly, suppose n to be odd and P n to contain only odd powers of 

 /A, then \P n dp. will contain even powers of /x with a constant term and will 

 be divisible by (l /u, 2 ). 



,7m n 



13. We have taken 



hence 



Vn 7 m \* p I . 



L( |X 

 P i r /' P 



* n / 1 n\ > tl/ f n 



