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) 2” = 2(2— 1) (GE 2) ae (cr —n-+ 1) n=l 203 aoe 
gh tes 
If we set 
(2) 2° =A(o, n)2+4 ACA, noi +..... + A(k, n)o™+ .... + Aan) o™ 
it is easily shown that 
(3) A(k, n) = S(k, n)/S(k; &) 
whence 
(A) erAl (ear) peak — 10), Seg De ete 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, n): 
n 
= s (n—1 : =" if 
(5) A(k,n) => ee AkK—1i—1) => 2 1725) Ae 
i=k i=k 
The recursion formula 
(6) A(k,n) =k A(k,n—1)+ A(kK—1,n—1) 
by which may be constructed 
A TABLE OF VALUFS OF A(k, n) 
k=0 k=1 k=2|k=3 | k=4 | k=5 | k=6 | k=7 ee | 
| in ee 
| n 0 | 1 | 
n= 1.| 0 1 
w= 2 (0) 1 1 
fae = 3 0 1 3 1 
| n=4 0 1 7 6 1 | 
n= 5 0 1 15 25 10 be) 
n = 6 0 1 | 31 | 90 65 fo.) tan | | 
| n=7 0 1 | 63 | 301 350 140 | 21 1 
| n= 8 | 0 1-27 | 966 1701 1050 | 266 | 28 1 
To any entry add the product of the one on its rizht and the value of k above the latter. 
**See for example Boole’s Finite Differences, Chap. IV. 
