OF THE FORM (p?-!)" d'"P n /dp m x dfP q /dff. 391 



Putting q for p + 1 we see that the coefficient of 



in (p, i -l) p n m P n xD p P, 1 (when q=p+l) is 



2p-2r+l)l (n + q-r)l 



_ 



' r\ (q-r)\ (n-r}\ (n-m)l (n + m-2r+l)\ (2n + 2q-2r+l)l 



x [(n + m - 2r + 1) 2q - 2r (2n - 2r + 1)]. 



NOTE. The quantity in square brackets in this expression 



= [(2n-2r+l) (2q - 2r) - 2q (n - m)] , 

 or [(n + m + l)2q-2r(2n + 2q-2r+l )] 



or [{(n + q- 2r) + (m -p)} 2q - 2r (2n - 2r + 1)] 



or [(n + q + m -p) 2q - 2r {2 (n + q) - 2r + l}] 



or [( + m + q -p 2r) (n m + q +p 2r+ l) (n 



Hence the coefficient of (2n + 2q-4r+ 1) D m -"P n+tl _ n _ r in 

 (^-l) p Z) m P, l xZ J> P 9 (when g-=^ + l) is 



(q +p) I (2n 2r) I (n + m) ! (n + q r) I (n m + q +p 2r) ! 

 ' r! (q-r}\ (n-r}\ (n-m)l (2n + 2q-2r+l)l (n + m + q-p-2r)\ 



X [(2n - 2r + 1 ) (2q - 2r) -(q+p+l)(n- m)] 



(n + m)l(n + q-r)l(n-m + q+p- 2r) ! (2q - 2r) ! 

 l > r \(q- r) \ (n - r) I (2n + 2q - 2r + 1 ) ! (n + m + q-p- 2r) ! 



r(2n-2r+l)\(q + p)l _ (2n-2r)l (q+p+ 1)!~| 

 \j2q-2r-l)l(n-m)l (2q-2r)\ (n-m- 1) ij ' 



This may also be put in the form 



_ (q+p)\(2n-2r)\(n + q-r)\ 



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



r(n-m + q+p- 2r+ 1)! (n + m)\ _ (n-m + q + p- 2r) \(n + m + 1 



p-2r-l)\(n-m)\ (n + m + q-p-2r)\ (n-m- 1)!J ' 



There are several other forms in which this system of factorials may 

 be arranged. 



