444 
are exactly sufficient to determine the constants. These two equations, 
therefore, completely characterize 
k 
S(k,n) = > 1) , i" 
i=0 
In like manner, the difference equation 
(17) A(kn+k)— B(1,k) A(k,n+k—1)+..... ++ Cas B(i,k) A(kn+k—1) 
+....+ (1) Bik) A(k,n) = 0 
and the conditions 
AU a1) a= 10 eile O ro) ee eg lhl (Ken) en 
wae (k) 
ae . i Se ae cee iy eu 1 v n 
completely characterize A (k,n) S(E,E) a 1) i} ? 
B(k,n) satisfies the difference equation of order 2k + 1, 
Gein Bi athe), ees Bien 428) © cee + Cay (ae 
Btk,n+2k+1—i1) +...... — Btk,n) = 0 
of which the characteristic equation is 
(r— 1+! = 9 
Whence B(k,n) is a polynomial of degree 2k in n, but the k + 1 obvious 
conditions 
BE, n)-=0, n= 0; 1, 2,3, -.-: 2. k= 1,9 Bk) =F 
are not sufficient to determine the constants. It is possible, however, by 
the successive application of the method of differences, since 
A Bik, n) = (n+1) B(k—1, n) 
to determine these constants for any particular value of k. 
Thus: 
B(1,n) = 4 (n+1)n 
1 
B(2,n) = DA (n+1)n(n—1) (8n+2) 
RGn) = a (sient) es 
etc., etc. 
