OF THE FORM d m P n /dp m x cFPy/dpf. 381 



Substituting these expressions, the coeflBcient of D m+l> P n+p _ 2r+l in the above 

 square brackets becomes 



(n-m-2r+l) (p-r+ 1) (2n + 2p-2r + 3)-(n + m + 2p- 2r + 2)r (2n-2r + 1) 

 = {(2n + 2p-4:r + 3) + 2r}(n-m-2r+l)(p-r+l) 



- {(2n + 2p-4r + 3)-2(p-r + 

 = (2n + 2p-4r+3){n(p-2r+l)-m(p + l)-(p 

 = (In + 2p - 4r + 3) {(n - m - 2i- + 1) (p + 1) - r (2n - 2r +1)} 

 2p-4r+3) {(n m 2r+l) (p-r+l)-r(n 



Hence the coefficient of (2w + 2p-4r+3) D m+p P n+] ,_ 2r+l in D m P n xD"P p+l i 



s 



,_ y_ j>! 1.3.5 ...(2p + l) 1 . 3. 5 ... (2n-2r-l) 



' r\(p-r+l}\~ 1.3.5 ... (2n + 2p-2r + 3) 



x {(n m 2r 



In the expression for D m P n x D p P p+1 , the coefficient of 

 D m+ *P n+p _, r+1 x (2n + 2p - 4r + 3) 



expressed in factorials is 



,_ r 2(2j?+l)!x(2ro-2r)! ( n +p-r+l)\ 

 I ,_ 



! ( n -r)\ (2n + 2p-2r + 3)l 



x [(n -m+l)(p+l)- r (2n + 2p- 2r + 3)] 



3)] 



= I - Y ~ \(2n- 2r) ! (n+p-r+l)! 



' r! (p - r +1)1 (n -r) ! (2n + 2p-2r + 3) ! 



_ , v 

 7 r! (g r)! (n r)l (2n + 2q 2r+ 1): 



Hence in the value of D m P n x D^P^ when q =p, the coefficient of 

 (2n + 2q - 4r + 1) D m+p P n+q _^ 



expressed in factorials is 



, _ l}r (q+p)\(2n-2r)l(n + q-r)l 



I r! (q-r)\ (n-r)l (2n + 2q-2r+l)l' 



