250 
[205 
205. 
NOTE ON THE SUMMATION OF A CERTAIN FACTORIAL 
EXPRESSION. 
[From the Philosophical Magazine, vol. xm. (1857), pp. 419—423.] 
Mr Kirkman some months ago communicated to me a formula for the double 
summation of a factorial expression, to which formula he had been led by his researches 
on the partition of polygons. The formula in a slightly altered form is as follows: viz. 
v v [x + y + 2] ?/ \x\ J [r 4- k — x — [r — 1 — _ 2Jc [r + k + 2] k [r] fc 
[y + l]^ 1 [k - yf-v [k- 1 -yf-'-v = r + 3 [¿ + l] fc+1 [£]* ’ 
the summation extending from x = 0 to x = r — 1, and y = 0 to y — k — 1. In the 
particular case when k = r, then all the terms of the series except those in which 
y = x vanish; and putting therefore k = r and y = x, and making a slight change in 
the form of the right-hand side, the formula becomes 
^ [2x + 2] x [2r — 2x] r ~ 1 ~ x _ [2r + l] 7 ' -2 
[¿e-t-lp :+1 [r — xY~ x [r — l] r_1 
the summation extending from x = 0 to x = r — 1. 
We have, in the notation of Gauss, [m] m == m (m — 1) ...2.1 = Ilm, and a factorial 
[m] w is expressed in terms of the function II by the formula [m] il = IIto 4-II (m — n). 
Write also 
Hi (m - £) = (m -1) (m - f).. . f, 
II 2m =2- m Ilmll^m —£), 
II (2m + 1) = 2 2m+1 II//¿II] (m + ; 
we have