BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 559 



indeed any simple analytic curve without a horizontal tangent and 

 with vertical asymptote, provided that | P(.r) [ 9^ along the curve. 



If the equality sign obtains in (49) it will be necessary to employ 

 a curve with asymptote not quite in the vertical and to employ half 

 planes not bounded by a vertical line. 



It is also possible to replace S(.r) by certain anormal Jorms,^^ and 

 thus extend the above results to the most general case. 



Our conclusion may be summed up as follows : There exists a linear 

 difference system (44) with matrix solutions Y~{x), Y+{x) ivhich either 

 possesses prescribed characteristic constants pjjj, Cij-^\ or else constants 

 Pj, rj + Ij, Cij^^^ lohere li, . . .,ln cire integers. For an arbitrary curve 

 which meets each line parallel to the real axis only once, having a ver- 

 tical asymptote, and which does not pass through a point \ P(x) \ = 0, 

 there exist such matrices Y-{x), Y+(x) with the further property that 

 I Y-{x) \9^ to the left of the curve while the elements of Y+(x) are ana- 

 lytic and I F+ (x) \ ^ to the right of the curve. 



It is worthy of note that this last property determines the location 

 of the poles of the elements of Y+(.v) completely: namely, they occur 

 to the left of the curve and at the points for which | P{x) \ = 0. This 

 appears from the formula 



Y-{x) = F-(.r)P-'(.r), 



which also permits us to affirm that the precise maximum order of pole 

 of any element of Y-^(x) is the order of the zero of | P{x) \. 



Part IV: The Linear ^'-Difference Equation Problem. 



§ IS. On Linear q-Difference Equations. 

 A linear g-difference system may be written 



(54) Yiqx) = Q{x) Y{x) \q\>h 



where Q(x) is a matrix of polynomials of degree ij. or less, in analogy 

 with the normal form (46) of linear difference systems. The apparently 

 more general case in which the elements of Q(.v) are rational in x may 

 be reduced to this form readily. Let the least common denominator 

 of the elements of Q{x), written as quotients of relatively prime poly- 

 nomials, be 



(.r — ai) . . . (.1- — ad 



29 Analogous to the anormal series for linear differential equations. These 

 forms have recently been obtained by Mr. P. M. Batchelder. 



