185 



- l(x 



|(x— Xm) 



The quantitj' a F(x) evidently satisfies the difference ecjuation of the 

 theorem, where 



F(x)^ _ _ 



(x-xl). ...!(x-xfO 

 X — 

 a F(x) also satisfies I since in the region defined the gamma functions in the 



x — 

 numerator are analj^tic and in the denominator are different from zero, a F(x) 



being a particular solution of the difference ecjuation, the general solution is 



F(x) = p(x)a'F(x), 

 where p(x) is an arbitrarj- periodic function of period 1. 



From the limit in Lemma I, §1, it is evident that I and II will be satisfied 

 if, and only if, p(x) is chosen so that it is analytic everj^where in the finite 

 plane and when x = ■>: , A < Rfx)< A + 1, satisfies the relation 



= b. 



where b is finite. This can be written 



f m — nl 



Z'= ± CC' 



p(x) e 



= h 



xlz'l — Qz' 



-Qz' 



(m — n)(- 



z + ^) — k 0+(m 

 e 



n) z+k 



= b. 



The use of Lemma II, §1, gives the form which p(x) must take to satisfy this 

 relation and thus completes the proof of the theorem. 



F(x) will in general be unicpiely determined if its value is known at cj+r+l 

 different points at wliich it is analytic. For then we should have a set of 

 q+r+1 ecjuations linear in the B's from which we could determine the con- 

 stants Bj. 



The form of the periodic function p(x) obtained by Mellin is 



[Ai Ap 1 

 h . . H I , 

 sin ^(x — Ci) sin Ti:(x — Cp)J 



where the c's are arbitrarj- with the exception that no two can differ by an 



integer. Mellin restricted a to be a real positive cjuantity and in case this 



is done the q and r of this paper become equal. In that case the identity of 



the periodic function of Mellin and the periodic function of this paper can be 



