434 



From (1) and (3) it follows that satisfies the linear difference equation 



in + l)/(n+ 1) + (n-k)f(n) = 



It is well known that the sum of the coefficients (x + y) 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 (x — y)" be multiplied 

 by c", (2c)", (3c)", (ftc) respectively; c being arbitrarily chosen dif- 

 ferent from zero, the sum of the products will vanish for n = 1, 2, 3, 



ft — 1 but not for n > k, e. g. 



k = 8 —8, 28, —56, 70, —56, 28, —8, 1 

 c = 2 2 n , 4", 6", 8", 10", 12", 14", 16 n 



The sum of the products vanishes for n = 1, 2 7; but not for 



n > 7; for n = 8 it is 10,321,920. 



(5) If the first k coefficients of (x — y) be multiplied term by term, 



with k n , (ft — 1)", (ft — 2)", I", (n, ft = 1, 2, 3, ) the sum 



of the products will be 



(— l)* +n if n ~< k and (ft + 1)! -1 if n = ft + 1; 



in particular 



,fc fft+11 ,, u k fft + 1] , .j fft + 2] t_i fft + 11 _ 



* — («— 1) I 1 , + (A — 2) i 2 !—■■■■ + (—1) I jfe — X J ~ X 



e. g. take ft = 5. 



1, —6, 15, —20, 15, 



-n ,n r,n n -.n 



0,4, 6 , 2 , 1 , 



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) ', (ft = 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 ^0, the sum of the products will vanish 

 if n<k; will be ( — d) (ft/) if n = ft; and if n = ft + 1 will be the product of this 

 last result and the sum of the terms of the arithmetic progression. 



. g. take ft = 6, d = — 1, a. p., 4, 3, 2, etc. 



1, —6, 15, —20, 15, —6, 1 



4", 3", 2", 1", 0", (-1)" (-2)" 



