BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 553 



associated properly equivalent canonical system has characteristic 

 constants only modified as stated. 

 Now we have 



I F(.r)| - \S(x)\ 



in the complete vicinity of x = oo , which is made up of the sectors 

 iTuT2), . . . (tn,tn+i)- It follows that the series for | ^5 (.r) | converges 

 and hence must be of the form 



eP,(x)+__ +Pn{x).^.i> (a-\ h . . • \ a9^ 0, 



\ X J 



so that always 



ri + . . . + Tn ^ p. 



But the above reductions increase ?"i + ... + r,j and do not affect 

 ] S(x) I or the value of p, and so must terminate. 



We can state then that ciihcr a solution of the stated problem, or of 

 a modified prohlem in which the constants ri, . . ., r„ of one of the singular 

 points are altered to ri + /i, . . . , /•„ + /„ respectively, where I., . . .,ln (i'''^ 

 integers, will exist. 



The matrix Y{x) thus obtained is not always unique. The most 

 general determination is however of the form P{x) Y{x) where P(.r) 

 is a matrix of polynomials of constant determinant which fulfills other 

 conditions. Thus the notion of "primitive systems" admits of 

 extension to the case of irregular singular points. ^^ 



Part III; The Linear Difference Equation Problem. 



§ IG. Formulation of the Problem. 

 Let 



(44) Y(x+ 1) = Q.(x)Y{x) 



be a linear difference system in which the elements of Q{x) are poly- 

 nomials of degree /j. in .r.^* In my earlier paper on linear difference 

 equations I demonstrated that, at least if the above ec^uation admits 

 a formal matrix solution 



(45), S (x) = [.i-"^ (pje-'^yx'-Jsij (x)], 



in which 5y (x) is a power series proceeding according to negative 

 powers of x with the determinant of the leading coefficients not zero, 



23 Cf. Plemelj, loc. cit. pp. 240-245. Like theorems may be proved in a 

 similar manner here. 



24 Essentially the most general linear difference system with rational coef- 

 ficients may be reduced to this form; see II, § 5. 



