THE GENERALIZED RIEMANN PROBLEM FOR LINEAR 



DIFFERENTIAL EQUATIONS AND THE ALLIED 



PROBLEMS FOR LINEAR DIFFERENCE AND 



g-DIFFERENCE EQUATIONS. 



By George D. Birkhoff. 

 Received June 9, 1913. 



The program of obtaining a characterization of a function in sim- 

 ple descriptive terms which are independent of the equations of 

 definition of the function is a famiUar one. To Riemann is due 

 the formulation of this characterization for the algebraic functions 

 and for the functions defined by ordinary linear differential equations 

 without irregular singular points. In both of these instances the 

 characterization involves a certain number of characteristic constants 

 — the monodromic group constants in the last mentioned instance. 

 Riemann also proposed the associated problem of assigning these 

 constants at pleasure.^ 



During the last few years I have discovered that the program 

 admits of extension in a number of directions. The aim of the present 

 paper is to solve the generalized problem of Riemann for ordinary 

 linear differential equations with irregular singular points, and the 

 analogous problem for linear difference equations and for linear g'-dif- 

 ference equations. The formulation of the first and second of these 

 problems has been given by me earlier.^ At about the same time as 

 myself, Norlund, in his fundamental work on linear difference equa- 

 tions, was led to formulate essentially the second problem.^ The 

 third is stated in the present paper. 



The problem of Riemann for linear differential equations in its 



1 Werke, (zweite Auflage) pp. 37-39, 67-69. 



2 Trans. Am. Math. Soc, 10, 436-470 (1909), and 12, 243-284 (1911). These 

 two papers will be referred to as I and II respective^. 



3 Memoires de I'Academie Rovale des Sciences et des Lettres de Danemark, 

 series 7, 6, 309-326 (1911); C. R. vol. 156, pp. 200-202 (1913). 



In the second of these papers, Norlund gives a formulation and explicit 

 solution of the hypergeometric difference equation problem. 



