181 



On Lixpi\r Difference Equations of the First Order With 

 Rational Coefficients. 



By Thos. E. Mason. 



This puiKT treats of the Ijchavior of tlie solutions of a first or<k'r linear 

 difference eciuation with rational eoeffieients as the variable ai)])roaehes 

 infinity in a strip i)arallel to the axis of iniaginaries. A unique characteriza- 

 tion of certain solutions is obtained to within the determination of a finite 

 number of constants. The same problem has been discussed by Mellin.* The 

 treatment here given is much shorter and simpler. The proof has been sim- 

 plified by making use of the asymptotic expansion for the gamma function 

 and by knimall found in §1 of this paper. The use of this lemma has also 

 permitted the removal of some restrictions made by Mellin. 



Carmichaelf has shown that certain solutions of the first order homo- 

 jieneous linear difference ecjuation are uniciuely characterized by their be- 

 havior as the variable approaches infinity in the positive or the negative 

 direction parallel to the axis of reals. 



§1. Lemmas. 

 Lemma L If x = z + \z\ x's = uj -f ivj, x'j = u'i + iv'j, (hcti 



lim 



\ (x-xi ) . . . I (x-xm ) (m-n) (-z + ^ )-k '\^ ^ ! z ' 

 z^ e 



2 



(X— Xi>) . . . |(x— x>n ) 



0-t-(m— n)z-f-k 

 e 



= c, 



where 



z' — vj z' — v'j n m 



Oj = tan-i , O'j = tan-' and O = i] v 'jO'j— S v.iOj , 



z — uj z — u'j j = l j = l 



and u'hercX 



n m 



k = SR (x'j) — 2R (xj). 

 j=l j=l 



*Acta Mathematica 15 (1891): 317-384. S:e §§1-3 of the pane.-. In §3 ot an article in Mathema- 

 tis^he .4nnalen 68 (1910): 30.5-337, Mellin has defined a function by means of the linear homogeneous 

 equation 



F(x-hl)— r(x)F(x)=0, 

 where r (xJ has the particular form 

 (x — xi) .... (x — xm) 



r(x) = ± — — 



(x— xU) . . . (x — x'n) 

 fTransastions of the American Mathematical Society 12 (1911): 99-134. 

 tR(x) is used to deno;e the real part of x. 



