384 ON THE PRODUCT OF ANY TWO LAPLACE'S COEFFICIENTS 



Each of the above quantities is included in the expression 



art ' D*P x D>P - 2 ( - 1 V - - 



P} ^" X ' rl(q-r)\(n-r)\(2n + 2q- 



x (2w + 2g - 4r + 

 -j?)! (2g - 2r) ! 



- 2?' - (q - 



--- ! 



i (g"P) ! ( 2n - 2r + g -V - g ) ! ( 2 g - 



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



From which it appears that the general term in the expansion of D m P n x D v P q 



)\ (2n-2r)\ (n + q-r)\ (g-p)\ (2g-2r)! 



q 



is 



_ _ 



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



x(2n + 2 ? -4r + l)D' + PP )1 + 4 _ 2 ,., 



where s takes all values from to (q p). 



Cancelling common terms in numerator and denominator we may reduce 

 this to 



8. Now let m and p 0, so as to reduce to the case of the 

 product P n x P q . 



Then the coefficient of (2n + '2q 4r+ 1) P n+q . N will become 



y (n + g-r)\(2q-2r)lnl _ (q + s)\(2n-2r + q^s)\ 



' rl(q-r)l(n-r)\(2n + 2q-2r+l)\ ' sl(q-8)\(n-s)\(q-2r+8)\' 



But by a former investigation (see p. 374) the coefficient" in this case is 



A(n-r)A(r)A(q-r) 



where A (r) x2 r x(r !) 2 = 2rl 



Hence this coefficient 



_ (2n - 2r) I 2r \ (2q-2r)l ((n + q- r) \)~ X 

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



