454 
From (1) and (3) it follows that bal satisfies the linear difference equation 
mp l)i@t 1) @w—k) fm) = 0 
It is well known that the sum of the coefficients (x + yy is 2° and that 
the sum of the odd numbered coefficients is equal to the sum of the even num- 
bered ones; the following are perhaps not so well known: 
(4) If, beginning with the second, the coefficients of (« — y) be multiplied 
bye 5 (20) 2 (BE) y esx (ke)" respectively; c being arbitrarily chosen dif- 
ferent from zero, the sum of the products will vanish for n = 1, 2, 3, ..... 
k—1 but not forn Ss k, e. g. 
k=8 —§8, 28, —56, 70, —56, 28, —8s, Il 
ee: a File 6". et 10", ee 14”, 16” 
The sum of the products vanishes forn =1,2...... 7; but not for 
n > 7; for n = 8 it is 10,321,920. 
(5) If the first k coefficients of (c= yee be multiphed term by term, 
with k", (k—1)", (K—2)",..... Dea lligen GAGE, coi aPe nous Tee ) the sum 
of the products will be 
k+n 
(—1) ii eon and (k + 1)!-1 Mee (eet 
in particular 
k (k+1) 
i . k (k 
Bip | —&—D 
co) er ae al 
e. g. take & = 5. 
1, —6, 15, —20, 10, 
The sum of the products is +1, —1, +1, —1, +1, 719, for n = 1, 2, 3, 4, 5, 
6, respectively. 
Both (4) and (5) are special cases of 
(6) If the coefficients of (x — y)*, (k = 1, 2, 3, ...) be multiplied term by 
term by the nth powers (n = 0, 1, 2, ...) of the terms of any arithmetic pro- 
gression with common difference d = 0, the sum of the products will vanish 
if n<k; will be cary (k!) if n = k; and if n = k + 1 will be the product of this 
last result and the sum of the terms of the arithmetic progression. 
.g. take & = 6, d = —1, a.p., 4, 3, 2, etc. 
V==6, 15 oo is —— 1 
Ar. 3", ae igs Or = 1 Ne (— Dy” 
