184 On a Theorem for the Development of a Factorial. 



through the notation first made use of. It is clear that the 

 above theorem includes the binomial theorem for positive inte- 

 gers, the corresponding theorem for an ordinary factorial, and a 

 variety of other theorems relating to combinations. Thus, for 

 instance, if C q (a x . . . a p ) denote the combinations of a x . . . a p> 

 q and q together without repetitions, and H <? (a 1 . . . a p ) denote 

 the combinations of a x . . . a p , q and q together with repetitions, 

 then making all the a's vanish, 



x—a x . . . x— a p =S qo { — )%(«! . . . a p )xP-i j 



and ,\ {z-a) p =S q P o (-) q C q {a, a . . . plures) a?-* 



=s/ (-)^W-<, 



the ordinary binomial theorem for a positive and integral index p. 

 So making all the a's vanish, 



x p =8 qo H q (* 1 . . .u p - q +i)x— cc x x—u 2 . . ,x—ct p - q . 



If m be any integer less than p, the coefficient of x m on the 

 right-hand side must vanish, i. e. we must have identically 



0=8, »-"(-)*0,_,_.(«, «,. . .« P - e )H,( ai «,. . . v. f+I ). 

 So also 



Cp- TO («! fl 2 ...« ; ,)=S^" m (-) 9 C p _ g - w (a 1 « 9 ,..« p _,)l ! " "^J - 



Suppose 



Aj = 0, fl 2 =l...flp=p — 1 ; a 1 = £,a 2 =^ — 1> ••• «;,=£— jo-fl, 

 then 



L «u «/> J * L <U i>-i J * m« lj 



a?—**! . . . x— a p - q = [x] p ~ q . 

 And hence 



M 



the binomial theorem for factorials. 



A preceding formula gives at once the theorem 



U,(0,l...p-g)= ip ^ qr , A"-^. 



It may be as well to remark, with reference to a demonstration 

 frequently given of the binomial theorem, that in whatever way 

 the binomial theorem is demonstrated for integer positive indices, 



[*+*]'=S t ^ t [*]'[*]' 



