444 



are exactly sufficient to determine the constants. These two equations, 

 therefore, completely characterize 



S(k,n) = 2 (-1)* [J] i n 



i=o *■ ; 



In like manner, the difference equation 



(17) A(k,n+k) — B(l,k) A(k,n+k— 1) + + (— l) 1 ' B(i,k) A(k,n+k— i) 



+ .... + (—l) k B(k,k) A(k,n) = 



and the conditions 



A(k, n) = 0, n = 1, 2, 3 k—V, A(k, k) = 1 



completely characterize A (k,n) = ^ 2 ( — l) 4 •] i n 



B(k,n) satisfies the difference equation of order 2k + 1, 



(2k + l) 



(18) B(k,n + 2k + l)— {^i 1 ) B(k,n + 2k) + + (-1) { 



B(k, n+2k+l— i) + — B(k, n) = 



of which the characteristic equation is 



(x — l) 2k+1 = 



Whence B(k, n) is a polynomial of degree 2k in n, but the A" + 1 obvious 

 conditions 



B(k, n) = 0, n = 0, 1, 2, 3, k — 1, B(k, k) = k' 



are not sufficient to determine the constants. It is possible, however, by 

 the successive application of the method of differences, since 



A B(k, n) = (n + 1) B(k — 1, n) 



to determine these constants for any particular value of A". 



Thus: 



B{\,n) = \ (n+l)n 



B{2,n) = ^ (n+l)n(n— 1) (3n+2) 



B(3,n) = ^ (n+l)V(n— 1) (n— 2) 

 etc., etc. 



