366 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



x (n m + 1 ) -, 1 " +1 + (n + m) j~m~ 



rt p .7m p ] 



'"' P' /7 m P' 



* i + i ** * n-i 



dp. dp 



d'"P d'"P' 



The coefficient of - ' within the large brackets is 



dp dp 



- (n m\~ (n m + 1 )~ + (n in) (n m+\ )~ (n + m+l) 

 2 v / v / v / \ / 



+ - (n m) (n m+ 1) (n + m) (n + m+ 1). 



Zt 



Unite half the middle term to each of the other two and this reduces to 

 n( n ~ m ) (n m+ I) \_(n m + I ) (n m + n + m + 1) 



LJ 



+ (n + m + 1) (n -m+l+ n + m)] = (n - m) (n -m+l)(n+l) (2n + 1). 



Also it may readily be seen that the coefficient of 

 r/"' P cT" P' <l m P r}"' P' 



' *n + i _< ' r n-i , t/ r n-\ u * n + i 



dp" 1 d/j." dp" dpi' 



is equal to identically. 



fl'"- P fjm pr 



And the coefficient of ^-f~ ^- is 

 - (n + m) (n + m + I) [(n + m) (n + m+ I +n m) + (n m) (n + m + n m+ I)] 



Hence the coefficient of 2 cos m<j> in (2n+l) Q n q is 



(n-m-l)! 



