356 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



(n + 2-r)(m + r+l) , (m + r + 3)(n- r) 



* ^' ; 



, 



or calling / (r) = -r . , t-- , 



(m TO) (m + n+ 1) 



we have X = i/;(r-l) and ^ = i//(r+l); 



Hence we can at once sum the series 



/(l)+/(3) + &c.+/(TO). 



For 



&c. = &c., 

 /() = ^r( 



Hence /(I) +/(3) + &c. +/() = ^ (0) <f> (0) - /r (w + 1 ) ^ (71 + l). 



In this case evidently \fi(n+l) vanishes and ^(0) = ^-^ - r-f-^ 



^ v y - 



Ac. +/(n) - r (-+I ^ ( ) 



(m ) (m + w+ 1) r 



1.3.5 ...(m-1) 1.3. 5... 





(m-n)(m + n+l) 2 . 4 . 6 ... (m + 2) 2 . 4 . 6 ... (n + 1) 



1 1.3.5... (TO-I) 1.3.5 ...TO 



~ (m-TO)(m + TO+l) 2 . 4 . 6 . . . m 2.4.6 ... (TO-!) 



the sum required, whence 



P m P n dn = ( - 1) * 7- -rA -77T 1 ' 3 '^"/ m ~ -V-fl' 5 /" n i\ ' ' ' ( 9 )' 



(m n) (ni + TO + 1 ) 2 . 4 . o ... m z.4.b...(TO 1) 



Next, suppose m to be odd and TO even, m being still the greater. 

 Assume f(r) and < (r) to be of the same forms as before in m, TO and 

 r. Then since m and TO are here changed as regards being even and odd, 



f(r) will now be unmeaning unless r be even, 

 and d> (r) will be unmeaning unless r be 



