443 
or, setting n—k for n 
n—k 
(15) > C1)'A(k, n—i) BG, k) = 0 
i=0 
E ' provided n>k = 0,1,2,3.... 
+ C41)'A(k, n—i) Bi, k) = 0 
i=0 
The two sums are equivalent since for i>k, B(i,k) vanishes and for 
i>n—k, A(k, n—?) vanishes. 
From (15) 
k 
A(kn) = = (—1)'*' A(k, n—i) B(i, k), n>k =0,1,2,.. 
i=1 
whence 
k 
B(kn) = > (-1)'*' Bi, n) A(n, n+i), n>k =0,1,2,.. 
i=l 
Solving for the successive A’s and B’s, and for brevity writing Ai, A» for 
A(n,n+1), A(n,n+2) ete., and Bi, Bo, for B(1,k), B(2, k) ete., 
A(k,k) = 1 
A (k,k-+1) = B, 
Ah k-=2) = BB, 
Aleks) = Bo — 28 Bs B; 
Aha) = By abe B, 2B Be — Bye Bs, 
Ate heb i= 8, — 4B! 8, 9B, 9B By B, 1-38. By— 2B iB, 
etc., etc. 
Bn) =1 
B(,n) = A, 
Bom == A, 
BEw a= A OAy AL SA 
ete., etc., in exactly the same form as the B’s. 
S(k,n) satisfies the linear difference equation of order k, 
(16) S(k,n-+-k) — B(1,k) S(k,nt+h—1) +... + 1)’ BG B) S(k,n+h—i)4+... 
_..+ (1)* Bik) S(k,n) = 0 
of which the characteristic equation has for roots 1, 2,3... . k; and the 
conditions 
S(k, n) = 0; n = 1, 2,3....4—1; S(k, &) = (—1)' k! 
