420 Mr. A. Cayley on the Summation of a 



T y l^ + y + ^y l^y [r + k-x-yf-'-' [r-l-x]"-'-^ 



U [r + ^ + 2]^ {rf 



"r+s [k+if-"' \_kY 



the summation extending from x=0 to x = r—\, and y=Q to 

 y = k—l. In the particular case when k = r, then all the terms 

 of the series except those in which y= a; 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 



[2a; + 2r [2r-2^]'-'-" ^^ [2r+l]'-^ 

 [a?+l]"+' \r-xY-' ['—!]'■"'' 



the summation extending from a?=0 to a^= r — 1. 



We have in the notationof Gauss [7/i]"' = ??j.m — 1 ...2.1 =!!»?, 

 and a factorial [in']"' is expressed in terms of the function 11 by 

 the formula [m]"=nm-H-n(»z— n). Write also 



ni(«-i) = (m-l)(7w-f)...|.l, 

 we have 



n2m=2""nmn,(OT-i) 



n(2m+l)=2""+'nmn,(m + i). 

 And transforming the factorials by these formulae, the series 

 becomes 



^ ni(^+i)n,(r-^-i) ^ 2rn,(>-+i) 



*" n(a; + 2)n(r-^ + l) n(/- + 3) ' 



the summation, as before, from x = Oiox=r — \. This may be 

 written 



ni(^+i) n,(/— ^-1) n2 n(r+i) ^ sk^+j) 



^ n,(i) ■ n,(r-i) Ti{x + 2)U{r-x + \) (r + 2)(r + 3)' 

 the summation from x = Q to x=r — \. The general term does 

 not vanish for x=7' or x = r + \, but it vanishes for all greater 

 values of x; hence if we add to the right-hand side the two 

 terms corresponding to ^ = r and x = r + \, the summation may 

 be extended from .r'=0 to x = r + \, or what is the same thing, 

 from ar=0 indefinitely. The two terms in question are 

 4(?- + ^) , -8(r + |)(r + l) _ -4r(r + A) 

 ?- + 2 ^ [r + 2)\r + Z) ~(r + 2)(r + 3)' 

 and the resulting equation is 



^ Ti,{x + ^)Tl{r-x-^) n2 n(r + l) ^ 4r(r + |) 

 ^ n^d) ni(r-l) n(a; + 2)n(/— ^+1) (r + 3)(r + 3)' 

 the summation from x=0 indefinitely; or substituting for the 



