OF THE FORM d m P n ldp. m ^d^P q ld^. 377 



TU J.-.L u i n , \ [( n ~ in)m + r(2n 2m 2r+ 1)1 



The quantity in brackets = (2n+ 1) * - 'r- *- ; 



r(2n 2m 2r + 1) 



hence 



uZ>'" +I P = S I (m + r-1)! 1.3. 5 ... (2n-2r-l) 



*\ m!r! 1 . 3 . 5 ... (2n-2m-2r+ 1) 



x (2n - 2m - 4r + 1) P M _ 2 ,.{(ft - m) (m + 2r) - r (2r - 1 



4. To express the value of (1 - ^) p D m+1> P n . 



We have (2n + 1) D m P n =D m+1 P n+l - D m+l P n _,. 



Also (2n + 1 ) /tD" +I P B = (n - m) D m+l P n+l + (n + m + 1 ) IT +l P n _ t . 



From the fundamental differential equation we get 



Hence 



(2ft + 1) (1 - ,r) Z)'^P )( + ( n _ TO ) ( w _ OT _ i) D m+l P 



n+l 



Or 



n ,,n7)"' +2 P- (^- 

 "**' 



Multiply by (1 /r) and repeat the above process on the right side of the 

 equation, then 



/ 1 2 \ 2 n'+2 p _ ( 

 ~ / *' 



_ 9 (ft - m) (ft - m - 1) (ft + m + 1) (n + m + 2) T)m p 

 (2ft -1) (2n + 3) 



(ft 



+ m + 2) (71 + m + 1 ) (n + m) (n + m - 1 ) r}m p 

 2n-l2ft+l 



(2n-l)(2ft+l) 

 By repeating the above process we get 



/ 1 fl ,\, n p . _ (ft-m + 3)(ft-m + 2) ... (n-m-2) ^ p 

 ^- (2+l)(2ft + 3)(2ft+5) 



2 "1 ,,, 

 -lJ " 



... __ _ 



__ _ 



(2ft+l)(2ft + 3) 2W + 5 2 



A. n. 48 



