Isomonodromy transformations of linear systems of difference equations. (English) Zbl 1085.39001

A general theory of isomonodromy transformations for a system of linear difference equations \(Y(z+1)= A(z) Y(z)\) with rational coefficients is developed. It is shown that if two equations with coefficients \(A(z)\) and \(\overline A(z)\) have the same monodromy matrix, then there exists a matrix \(R(z)\) such that \(\overline A(z)= R(z+ 1)A(z) R^{-1}(z)\), where the characteristic constants \(\{C^{(s)}_{kl}\}\) are the same while the constants \(d_k\) are being shifted by \(sk\in z\). Finally, it is proved that Schlesinger difference equations and Schlesinger transformations can be obtained as limits in two different limit regimes.


39A05 General theory of difference equations
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