358 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



m+1 __1 



7 m(m-n+ 1) (m + n + 2) (m-n-l)(m + n) 



2m + 1 1 _1 



~ m(m n+ 1) (m + n + 2) (m nl)(m + n) (m n+ 1) (m + n + 2) 



2m +1 



m(m n+l) (m + n + 2) (m-n l)(m + n)(m n+l)(m + n + 2) 



_/ 2m n f (m-n-l)(m + n) + 2m \ 



' \m (m-n+1) (m + n + 2) (m - n - 1) (m + )/ 



_ _ (2m + 1 ) (m - n) (m + n + 1 ) _ 

 ~ m (m n + 1) (TO + ?i + 2) (m n - 1 ) (m + n) ' 





p 

 5 J o ^,, f B rtf* = 



2 ) 



1.3.5 ... (m-1) 1.3.5 ...(n-1 ) 

 2. 4. 6.. .m. 2. 4. 6.. .n 



P,,^ .................................................. (11). 



Next suppose m and n to be both odd and m to be greater than n. 



(\p m c^ 



Jo 



( -_lp 1.3.5 ...m. 1.3.5 ...(n-2) 

 (2n+ 1) 2.4.6... (j - 1) 2 . 4 . 6 ... (n+ 1) 



( n n+l 1 



C \(m-n- 1) (m + n + 2) (m-n+l)(m + n)j ' 



n TC + 1 



Now 



(m n 1) (m + n + 2) (m 



m(m+l)-n(n+l)-2 

 Hence 



(m n 1) (m + n + 2) (m n+ l)(m + n) 



1.3.5 ... m. 1.3.5 ...(n-2) , . 



7 >< 7 77H i \ o ^ Z 7^ t i \ V / 



