554 PROCEEDINGS OF THE AMERICAN ACADEMY. 



there exist two matrix solutions F~ (x) and Y+ (.r), with elements 

 analytic in the finite plane save for poles, such that Y~ (x) <^ S (.r) in 

 any left half plane and Y+ (x) o^ S (x) in any right half plane. ^^ The 

 existence of such a solution was proved by Norlund and Galbrun by 

 methods based on the Laplace transformation somewhat earlier.^ ^ 

 These matrices Y~ (x) and 1 + (.r) are connected by a relation 



(4G) Y- (.!•) = }'+ (•>•) P (x) 



where P (x) is evidently a matrix of periodic functions of period 1. 



From the form of (44) it appears that Y^ (x) is a matrix of entire 

 functions, while 1 + (.r) is analytic save for poles. 



In my paper I determined explicitly the nature of the elements 

 Pij (x) of P (x) to be the following: 



(47) 

 ' pa (x) = 1 + Cu(i)e2- ^-1^+ . . . + Ciie2-(M-i) V-ix^ ^/2^ri V-ig2^M ^l-^x 



(i = 1, . . ., n) 

 Pij{x) = e'-^^ii ^^^[cyW + . . . + Cy2-(''-i) ^^'^1 



(i ^ i; ij = 1, . . ., n), 



Here X,j stands for the least integer as great as 



(48) 9t(;^-^[l0gp,-l0gpi]). 



An analogous determination in certain cases at about the same time 

 was made by Norlund (loc. cit.). 



It is not difficult to show that, if Y~ (x) and }'+(.v) have the properties 

 above outlined, then conversely they are solutions of a linear difference 

 system (44) in which the elements of Q (x) are rational if not polyno- 

 mial.^^ For this reason the constants pj, Vj, Cij^'^'^ may be called the 

 characteristic constants of Y~ (.r) and Y+ (.r). 



This characterization suggested to me the following problem: To 

 construct a linear dift'erence system (44) with assigned characteristic 

 constants in which the elements of Q (.r) are polynomials in x of 

 degree not greater than /x. 



25 These matrices Y-{x) and F+(x) correspond to G^r) and //(x) of II. In 

 certain cases it may be necessary to consider half planes not bounded by a 

 vertical line. I refer to this possibility later. 



26 Norlund, Dissertation, Copenhagen (1911); Galbrun, Dissertation, Paris 

 (1910). 



27 See II, § 7. 



