181 



On Linear Difference Equations of the First Order With 

 Rational Coefficients. 



By Thos. E. Mason. 



This paper treats of the behavior of the sohitions of a first order linear 

 difference ecjuation witli rational coefficients as the variable ajiiiroaches 

 infinity in a strip parallel to the axis of imaginaries. A unique characteriza- 

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

 number of constants. The same problem has been discussed by INIellin.* 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 Itmnia II fovmd 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- 

 geneous linear difference equation are unifjuely characterized by their be- 

 havior as the variable approaches infinitv in the i)ositive or the negative 

 direction parallel to the axis of reals. 



§1. Lemmas. 

 Lemma I. If x = z + iz', xj = uj + ivj, x'j = u'l + iv'j, then 



lim 

 z' 



|(x— xi ) . . . l(x— xna ) (m— n)(— z + A)— k - 

 z' e 



(X-Xi') . . . |(x-x'„ ) 



0+(m— n)z+k 

 e 



= c, 



where 



z' — vj z' — v'j n m 



Oj = tan-' , 0'j = tan-' and = ^: v'jO>j— iSviOj, 



z— uj z— u'j j = l j = l 



and uiieret 



n m 



k = 2R (x'j)— :SR (xj). 



j=l j=l 



*Acta Mathematica 15 (1891): -317-384. S;e §§1-3 of the paper. In §3 of an article in Mathenia- 

 tis?he Annalen 68 (1910): 305-337, 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— x'l) . . . (x — -x'n) 



tTransa?tions of the American Mathematical Societj' 12 (1911): 99-134. 

 }R(x) is U:ed to denote the real pa^t pf x. 



