BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 565 



The relation (68) with the expUcit values of yoU), J/^jCi') and p (x) 

 substituted in is essentially one of the fundamental product formulas 

 for the sigma function. 



§ 20. The q-Difference Equation Problem. 



It is now easy to show that conversely if }'o (-^O, i'oo (-^'^ ^^'^ matrices 

 of functions of the form (58) in the vicinity of x = and a- = co 

 respectively, analytic for .r ?^ 0, go , save for poles, and if the matrix 

 P{x), defined by the relation \\{x) = Y^(x)P (x), is composed of 

 elements Pij(x) which are left unchanged when .r is replaced by qx, 

 then Yq (x) and Y^^ (x) are matrix solutions of a linear g-difference 

 system (54) with rational coefficients. In fact, if we write 



it is seen at once that for x 9^ 0, cc, the only singularities of Q{x) 

 are poles, while the first and second of these forms for Q(x) ensure that 

 Q{x) is composed of elements analytic at x = and with a pole of at 

 most order ^t at x = 00 if not analytic there. This suffices to establish 

 the fact that the elements of Q(x) are rational.^* In order to conclude 

 further that the elements of QXx) are polynomials of degree fj. it is 

 sufficient to know that the plane may be di\'ided into two parts by a 

 loop about X = meeting each equiangular spiral or radial line 



(70) 6 = c -{- , — ^-^ log r (r, 6, polar coordinates) 



log I q \ 



only once and not passing through a point | P{x)\ — 0, such that the 

 elements of Yq{x) are analytic and |ro(-^')| is not zero within or along 

 the loop, M'hile the elements of Y^ {x) are analytic and | Y^ (.r) | is 

 not zero outside the loop. Under these conditions the first expression 

 for Q (.r) makes it evident that its elements are analytic within or 

 along the loop, and the second expression makes it clear that the same 

 is true without the loop, since if x is a point without the loop so is qx. 

 It is natural to term the 2n constants pj, aj and the ?r (/x + 1) con- 

 stants Cij, ai'^'^\ . . ., aj-''-'^ the characteristic constants. These constants 

 are not all independent, since there are n^ relations between the con- 

 stants afc^*'-'^ of the type (67). Furthermore the constants c,j are not 

 uniquely determined by the given g-difference system. For, any ith 

 column of }'o (.v) is only determined up to a constant factor j) and like- 



34 Compare II, § 7. 



