Second-order functional-difference equations. I: Method of the Riemann-Hilbert problem on Riemann surfaces. (English) Zbl 1064.39016

The authors propose a new analytical method for scalar second-order functional-difference equations with meromorphic periodic coefficients. The technique involves reformulating the equations as a vector functional-difference equation of the first order and reducing it to a scalar Riemann-Hilbert problem for two finite segments on a hyperelliptic surface. The method is applied to solve in closed form a second-order functional-difference equation when the corresponding surface is a torus.
[For part II see ibid. 57, No. 2, 267–313 (2004; Zbl 1064.39017), reviewed below.]


39A12 Discrete version of topics in analysis
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable


Zbl 1064.39017
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