OF THE FORM d m P n /dp m x dfPJdf/f. 389 



Hence using this last expression the coefficient of 



(2n + 2q - 

 in 2lTP n xD v P q will be 



T\ (q r)l (n r}\ (2n + 2o 2r+ 1)! 



\-//\ / \ i / 



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

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



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

 (n m 2r + 1 ) ! (n + m + 2q 2r 2) I (n + m 1 ) ! (n m) I 



(n + m - 2) I (n- m) ! 



_ , . ,. (q +p) I (2n 2r) ! (n + q r) ! (n m + q p 2r) I (n + m) ! 



' r] (<?-~r)! (n-r)\ (2n + 2q-2r+ 1)! (n + m + q+p-2r)\ (n-m}\ 



+ m + 2(/ - 2r) ! (n - m) ! (n + m + 2q- 2r- l)! (n m+l)\ 



- 2 7 



(n m 2r) I (n + m) I (n m 2r + I ) ! (n + m 1) ! 



(n + m + q+p- 2r) I (n - m + 2) !"| 

 (n - m + q -p - 2r) I (n + m - 2) ! J ' 



Taking the first of the expressions in the above note, the coefficient of 

 2q-4r+l)D m+1 'P n+rj ^ r in 2D m P n .DP^ will be 



(q+p)l (2n 2r)l (n + q-r)l 2r\ (n m + qp 2r)l 

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



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



\_2rl (n-m-2r)\ (2r-l)\ (n-m-2r+l)\ (2r-2)\ (n-m-2r-l 



_, . r 2r ! (n + q r) ! (n m + q p 2r) ! 

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



r 2^!(2n-2r)! (2q- 1)! (2n-2r+ l)\ 



X |_2r! (n-m-2r)\ (2r- 1)1 (n-m-2r+ 1)1 



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

 f (2r-2)! (n-m-2r + 2) 



The general term arising from t/his square bracket is 



/ \ ( ( 1~PY (2^ 6-)! (2n 2r + s)\ 

 ' s\ (qp s)\ (2r s)\ (n m 2r + s)l' 



where s has all values from to q p. 



