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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.

MSC:

39A05 General theory of difference equations
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