184 



Therefore we can write 



q : q 2djx 



pfx) = S Bj\aJ = i: Bje 



j=-r j=-r 



t Q . 



where q is the greatest integer < and r is the greatest integer 



= 2 2t: 

 t Q 



< 1 . From tlie definition of Q given in §2 the vahies of (j and r will 



= 2 2z 



not differ by more than 1 in the prohlciii of this i)aper. 



The converse is o})vious. 



§2. Homogeneous Equations. 

 Theorem. Every Jimt order linear homogeneous difference eq mil ion rrith 

 rnlional coefficients, as 



F(x + 1) - r(x) F(x) = 0, 

 ivhere r(x) can he irritten in the form 



(\ — Xi) . . (X — Xm) 



r(x) = a . a = he 'Q, — t.<Q<t., 



(x — X',) . . (X — X'n) 

 has a solution Fix) irhich has the following properties, provided that each of the 



m — n (2 



numl)ers ± is greater than zero, or in case m = n that Q = and 



4 2x 



n m 



k = S R(x'j) — >: R(xj) < 0. 

 j=l j=l 



I. F(x) is analytic in the finite [tart of the \-ptane defined liy R(x)>D, 

 where D is the greatest among the real parts of xi. x.. xm. 



II. ^.s x approaches infinity in the strip parallel to the axis of imaginaries 

 defined by A<R(x)<A + l (A>D) the absolute value o/F(x) remains finite. 



Every such function F(x) can he wrillen in the form 



|(x — x,) . . . . 1(X— Xm) c| 2djx 



Frx) = a'' — — X Bje 



(x — x'J .... ;(x — X,',) j= — r 



m — n Q 



where ci is the qreatest integer* < iiiid v the f/rrnlesl inlerger* 



4 27C ■ 



m — n (^ 



< + . 



4 2- 



m — n Q 



*The inequality sign should be replaced by the equality sign in ca^e each quantity =i= — 



4 2x 



is :»n intereer and at ihe same time th? exponent c.[ z' in the expression in lemma I, §1, is > ,that is 



when (m — .n) (— z +1 /2)— k> tor all valje-i of x in the s: rip define 1 in c )n(lition II nf the theorem. 



