443 



or, setting n — k for n 



n — k 



(15) 2 (— lfAQc, n—i) B{%, k) = 



_„1 



k 



2 (— I)*A(*, n—i) B(i, k) = 



i=0 



^ provided n>A; = 0, 1, 2, 3. 



The two sums are equivalent since for i>k, B(i, k) vanishes and for 

 i>n — k, A(k, n — i) vanishes. 



From (15) 



_ y 



A(k,n) = Z (—iy""A(k, n—i) B{i, k), n>k = 0, 1, 2, 



{=1 



whence 



B(k,n) = 2 (— l) 1+i B(k— i, n) A(n, n+i), n>k = 0, 1, 2, . . . 



Solving for the successive A's and B's, and for brevity writing Ai, A 2 for 

 A(n,n+1), A(n,n+2) etc., and B,, B 2 , for 5(1, k), B(2,k) etc., 



A (k,k) 



= 1 



A(k,k+l) = B, 

 A(k,k+2) = Bi — Bo 

 A(k,k+3) = B\ —2B 1 Bn + B 3 

 A(k,k+4) = B\ — 3B\b 2 + 2B X B Z — B, + b\ 

 A(k,k+5) = B\ — 4B\B 2 + w\B i — 2B x B i + B i +W,B\—2B 2 B Z 

 etc., etc. 



B(0,n) 

 B(l,n) 

 B(2,n) 

 B(S,n) 

 etc., etc., in exactly the same form as the B's. 



= 1 



= A x 



= A\ — A 2 



= A\ — 2A,A 2 + A 3 



S(k,n) satisfies the linear difference equation of order k, 



(16) S(k,n+k) — B(l,k) S(k,n+k—l) + . . . + (— 1)* B{i,k) S(k,n+k—i) + . . . 



. . .+ (—l) k B(k,k) S(k,n) = 



of which the characteristic equation has for roots 1, 2, 3 ... . k; and the 

 conditions 



S(k, n) = 0; n = 1, 2, 3. . . .k—l; S(k, k) = (— l) h k! 



