186 



2xix 



shown by making the transformation \v = e and equating the coefficients of 



like powers of w in the two transformed expressions for p(x). This gives a 

 system of p linear ecjuations to determine Ai, A>, . . ., Ap, where the p of 

 Mellin's paper is equal 2q + l. 



§3. Xf)N-HOMo(;E.\Eous Equations. 

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



fx — X,) ... . (x — xm) (x — X,") . . (x — x"g) 



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



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



where m > n, then the series 



X s(x+t) 



S(x) = 2 



t = o r(x+t) r(x+t — 1) ... r(x) 

 is always wn if or inly convergent for \a[>l and for |al = l when m>n, and is uni- 

 formly convergent for !a =1, m = n, when k — (g — n)>l, where 



n ni 



k = :£ R(xj') — :^ R(xj). 

 j=l j=l 



// the conditions for the uniform convergence of 8(x) are fulfilled, then every 

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



F(x+1) — r(x)F(x) = s(x), 

 has a solution F(x) which has the foUoiring 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 parts of Xi, Xj, .... xm. 



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

 A<R(x)<.\ + 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 — Xi) |(x — xm) q 2T:ijx «: s(x+t) 



F(x) = &""— = ^ Bje 



l(x — x>,) Kx— x'n) j = -r t = o r(x+t)r(x+t — 1). .r(x) 



r and q being defined as in the theorem in §2. 

 In the equation 



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

 make the substitution 



F(x)=f(x)u(x), 

 where f(x) is the solution of the homogeneous equation given in the theorem 



*If r(x) and s(x) do not already have a common denominator t\tc\- cMn c i^ily lie re lu mmI to ex- 

 pressions with a common denominator. 



