441 



Also 



k 



2 P + 1 ) Si= **(n + l) fc+1 -l 

 i=0 ^ j 



Relations between the A's and the B's: 



c m = 2 A(i, to) x {i) to = 1,2, 3 



i— 1 

 z W = 2 (— l) j S(i, i — l)^' - - 7 i = 1, 2, 3, 



Therefore 



m i — 1 



x m = 2 A(i, to) 2 (—1)^0', i — 1) x i_i 

 the coefficient of x" on the right is 



m — k 



2 (—1)' A (A + i, m) B(i, Jfc + i— 1) 



t=0 

 and this must vanish A: = 1, 2, 3, to — 1, and be equal to 1, for k = m. 



Whence, setting re for to — A, 



v f r* = o, i, 2, . . . 



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



i=o in = 1, 2, 3, . . . 



or, setting i for A + i, and re for to, 



(12) 2 (— 1)* A(t» 5(i— A, i— 1) = 0. | 



A = 0, 1, 2, re— 1 



re = 1, 2, 3, 



Similarly, starting from 



m — 1 



X = Z ( — 1) B{%, TO — 1) X 



1=0 



we obtain 



(13) 2 (— 1)* A (A, A+re— i) B(i, k+n— 1) = 0, = °' J"' *' ' ' 



i=0 (re = 1, 2, 3, . . 



This relation may be generalized as follows: 

 Set 



n 



C(k,n,p) = 2 (—1)*' A(k, k+n— i) B(i, k+n—p) 



i=0 

 **Prestet, Elements de Mathcmatique, p. 178. 



