185 



Th^ quantity a F(x) evidently satisfies the difference equation of the 

 theorem, where 



F(x) = 



l(x-x,) 



|(x— Xm) 



Kx-xl) 



(x-xf.) 



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



X — 

 numerator are analytic and in the denominator are different from zero, a F(x) 



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



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

 where p(x) is an arbitrary periodic fun(;tion 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 everywhere in the finite 

 plane and when x = oc , A < R(x)< A + 1, satisfies the relation 



a p(x) 



z'^ ± CO m — n 



(m — n)(— z + ^) — k 0+(m — n)z + k x|z' 



1 e e 2 



[I x| z -Qz' 

 where b is finite. This can be written a I = h c 



b, 



= b. 



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

 relation and thus completes the proof of the theorem. 



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

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

 q+r-(-l equations 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 . . +-_ I , 

 sin x(x — Ci) sin x(x — cp ) J 



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



integer. Mellin restricted a to be a real positive quantity 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 



•itj to 1 1 



