180 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



if n is any positive integer ; in consequence of this definition, 



(34) 



y" 



r 



y — n 



for 1 ^'n; we take the last formula as the basis of the definition of y"' 

 for n = 0, — 1, — 2, — 3, . . . , successively, and thus obtain 



1 



(35) y«>=.i, y-") 



(y+ l)(2/+2)(y + 3)... (5/ + n) 



for any positive integer n (that is, for any negative integer — n). With 

 this notation we have 



fy-' = y' • iy - ir^' = (- 1)" • (- y + n - 1)"", 



(y + « - 1)<«) == (- 1)« • (- y)'") 



(36) 



«"" = n : . 



,j(/i 



(« - ! 



where each of the letters n and i represents any positive integer or ; 

 also, 



j- (2 i) : = 22' • i ! {i - ^)''» 



\(2z + l)! = 2^'+^-i!(i + i)<'+^ 



(37) 



because, evidently, 2' i ! is the product of the even numbers from 2 to 2 i, 

 2' (i — i)''* is the product of the odd numbers from 1 to 2 ^ — 1, and 

 2'''"^ (i + 2)*'^^' is the product of the odd numbers from 1 to 2 i -\- I. 

 Furthermore, we have 



(y - 1)'"' ^(y- 1)"*-^' • (y - ^0 - y" ~n-(y- i)'-^' 



y" — n • y" 



+ m'^' • y 



„(3) . ^'H-5) _^ 



(38) 





where « is any positive integer or 0. 



From (32) it is evident that any polynomial in 7" changes p with any 

 even suffix into a linear function of the /)'s with even sulfixes and p with 



