438 



§3. 



In finding the sum of certain series by the method of differences** it 

 is convenient to express positive integral powers of x in terms of the poly- 

 nomials 



(1) x (n} = x(x — 1) (x — 2) (x — n + 1) n = 1, 2, 3, ... . 



,r (n) = 1 



If we set 



(2) x 11 = A(o, n)x W + A(l, n)x {V '+ + A(k, n)x (k) + + A(n,n) x {n) 



it is easily shown that 



(3) A(k, n) = S(k, n)/S{k, k) 

 whence 



(4) A(k, n), k, n = 0, 1, 2, 3, vanishes if n<k; is always positive if 



n > k > 0; in particular A(n, n) = 1; and the following relations come from 

 those given in §2 for S(k, ri): 



(5) A (k, n) = X \jZ i) A (k — 1, i — 1) = i 2 (,- ! jj il (k — 1, i — 1) 



The recursion formula 



(6) A (k, n) = k A (k, n — i) + A (A- — 1, n — 1) 

 by which may be constructed 



A TABLE OF VALTJF S OF A (k\ n) 



k = k = l 



'k = 2 



k = d 



&=4 



k = 5 



k = Q 



k = 7 



&=8 



n = 



1 



















n = 1 























n = 2 









1 















n = 3 









3 



1 













n = 4 









7 



6 



1 











7! = 5 









15 



25 



10 



1 









w = 6 









31 



90 



65 



15 



1 







» = 7 









63 



301 



350 



140 



21 



1 





n = 8 









127 



966 



1701 



1050 



266 



28 



1 



To any entry add the product of the one on its right and the value of k above the latter. 



"See for example Boole's Finite Differences, Chap. IV. 



