390 ON THE PRODUCT OF ANY TWO LAPLACE'S COEFFICIENTS 



11. Again from the expression for (/r l) p D m P n we may find the 

 value of (jL*-\yD n PxD v P v in terms of the form D*-*P, multiplied by 



constants. 



Thus from Art. 4 (p. 379) we have (^ - l)'D M P n x D V P P 



* r p\ 2". 1 . 3. 5 ...('2p 1)! (2n 2r)\ (n + m)\ (n m + 9,p 2r)\ (n+p r)l 

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



x (2n + 2p - 4r + 1) Z) m " p P n+J) _ 2T 



_ 2p I (2n - 2r) ! (n + m) I (n - m + 2p - 2r) I (n 4 p - r) ! 



) r i (p _ r ) t ( n _ r ) I (n - m) '. (n + m - 2r) I (2n + 2p - 2r + 1) ! 



Now multiply by (2p+l)ji/. and we get 



x (ft' lyD^P,, x D p P p+1 = terms of the form 



m)! (n m + 2f> 2r + 2)\ (n+p r+ 1)! 



v -! j3 - 



(r-l)l(p-r+l)\(n- 



x [(n -m + 2p- 2r + 3) Z)'"- J 'P )1+1 ,_, r+3 + (n + m-2r + 2) D m - 1 'P n+p _* +l '] 



. r (2p + 1) ! (2n - 2r) I (n + m) ! (n -m+2p- 2r) I (n+p-r)l 

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



x [( -m + 2p - 2r + 1) D"^P n+p _^ + (n + m - 2r) D m -*P n+p _ tr _ l '}. 

 Taking only the coefficient of D m ~ p P n ^ p _ zr+l in the result we have 



. r (2p+l)\ (2n-2r)\ (n + m)l (n-m + 2p-2r+l)\(n+p-r+l)l 

 I r \ (p_ r+1 )t ( n -r)\ (n-m)l (n + m-2r+l)\ (2n + 2p-2r + 3)\ 



x[-2r(2w-2r+l)(n-ra+2p-2r+2) 



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



The quantity in square brackets is 



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



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



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



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



