186 



shown by iii;ikiii>i the tnuisfoniKition \v = (' jintl ccnuitiiifz: tlic cocfliciciits of 



like powers of w in flie two transformed expressions for i)(x). This p;ives ii 



system of p linear equations to determine Ai, A-.. Ap. wlierc tlie p of 



Mcllin's paper is equal 2ci + l. 



§3. Non-homogeneous Equations. 

 Theorem. //r(x) and s(x) are rulional functions of the form* 



(x — X,) (x — xm) (x — x,i') . . (x — x"g) 



r(x) = a , s(x) = b , 



(x — x'l) ... (x — x'n) (x — X',) . . . (x — x>n) 



where m > n, then the series 



00 s(x+t) 



S(x) = S 



t = o r(x+t)r(x+t — 1) . . . r(x) 

 IS ahv<ii/n uniformly convergent for [a|>l and for |a| = l when m>n, and is uni- 

 fornilij convergent for !a[ = l, m = n, ivhcn k — (g — n)>l, where 



n m 



k = S R(xjM — S R(xj). 

 j=l j=l 



// the conditions for the uniform convergence of S(x) are fulJiUed, then every 

 first order linear non-homogeneous difference equation with rational coefficients, as 



F(x+l)-r(x)F(x) = s(x), 

 has a solution F(x) which has the following properties: 



I. F(x) is analytic in the part of the finite x-plane defined by R(x)>D, where 

 D is the greatest among the real j)arts of x^ x,, .... xm. 



II. // X is confined to the strip parallel to the axis of imaginaries defined by 

 A<Ii(x)<A + l (A>D) the absolute value of F(x) remains finite as x approaches 



infinity. 



Every such solution F(x) can be written in the form 



|(x — X,) Kx — xm) (1 L'-ijx X s(x+t) 



F(x) = a'=r= — l'15.ie 



|(x — xVO j(x— x'n) j= r t = o r(x + t)r(x+t — 1) . .r(x) 



r (iml (| being defined «s in the theorem in §2. 

 In t he ('(luation 



F(x+l)-r(x)F(x) = s(x) 

 make the substitution 



F(x)=f(x)u(x). 

 where f(x) is the solulion of (lie homou;eni'ous e(|uati()n nixcii in the llieorem 



*If r(x) and s(x) (!:> not iilri-ady liavo a coiiiMinii <lc!iimituilor tlio.\- ran o i>il.\- \k- re In -cil lo ox- 

 prcssion.s with a common denominator. 



