184 



'riicict'oi'c we can write 



q ci 2-ijx 



p(x) = ^ Bjw^ = 1' Hiv 



whcM'c q is th(> greatest intcsier < and r is the greatest integer 



= 9 2r. 



t Q 

 < 1 . From the ilefiiiition of (^ given in i!2 the vahies of i[ and r will 



= 2 2x 



not differ by more than 1 in the ])i-()blem of this paper. 

 'I"he converse is obvious. 



§2. Homogeneous Equ.\tions. 

 Theorem. Every JirsI order linear hnniogrnroii^ differrnrr cqiintUm vHh 

 raiiondl coefficients, as 



F(x + 1) — r(x) F(x) = 0, 



irhere r(\) ran he written in the form 



(X — xi) . . (x — xm) 

 r(x) = a , a = he iQ, — t:<Q<x, 



(x — xS) . . (X — X'n) 



has a solution F(x) which has the following i)roperties, proriited that each of the 



m — n Q 



nuniljers ± is qreater than zero, or in ca--^e ni = n that Q = and 



4 2x 



n m 



k = 2 R(x'j) — 2 R(x.i) < 0. 

 j=l j=l 



I. F('x) ;'-s analytic in the finite part of the x-j)lane defined l)y R(x)>D, 

 icherc D is the (jreatest among the real parts of Xi. x.>, .... xm. 



II. Ak X approaches infinity in the stri/) parallel to the axis of imaginaries 

 defined t)y A<R(x)<A+l (.\>D) the aljsoliile raliie of F(x) remains finite. 



Erery sach function F(x) can be irritten in the form 



i(x — Xi) . . . . i(x— Xm) C) 2T:ijx 



F(x) = :i'' — ^ i: Hie 



(X — x'J . . . . |(x — X,',) j= ~ r 



m — n C2 



where ({ is the (/realesi integer* < ami r the (/reatest interger* 



1 2x 



m — n (.1 



< + — . 



4 2t. 



m — n Q 



*Tlu' inc(nialil V sisn sluiijld hi' icplar-ivl In I l.c (MHialii v .sinn in ca.'C cacli (|ii:iiuily =F — 



4 2- 



i-i :in intcriror and at the sumo time tlr.' fxponeril ( f z' in I lie oxpivs.sion in li'imiiM I. SI. i.s > .that is 



wlien (in— n) (— /, + 1/2)— k> lor iill valui'-i "f x in llic s. rip define I in c indiii.m 1 1 cif I lie theorem. 



