We introduce and study isomonodromy transformations of the matrix linear difference equation Y(z+1)=A(z)Y(z) with polynomial (or rational) A(z). Our main result is a construction of an isomonodromy action of Z^{m(n+1)-1} on the space of coefficients A(z) (here m is the size of matrices and n is the degree of A(z)). The (birational) action of certain rank n subgroups can be described by difference analogs of the classical Schlesinger equations, and we prove that for generic initial conditions these difference Schlesinger equations have a unique solution. We also show that both the classical Schlesinger equations and the Schlesinger transformations known in the isomonodromy theory, can be obtained as limits of our action in two different limit regimes.
Similarly to the continuous case, for m=n=2 the difference Schlesinger equations and their q-analogs yield discrete Painleve equations; examples include dPII, dPIV, dPV, and q-PVI.
On two Thomae-type transformations for hypergeometric series with integral parameter differences
本文对r对数值参数和计数参数相差正整数的超几何级数，得到了两个新的双曲型变换。这是通过应用由Krattenthaler和Rao [Symposium on Symmetries in Science（ed. B. Gruber），Kluwer（2004）]到最近得到的两个欧拉型变换。给出了一些特殊情况。
In this short paper, we show how to deduce several types of generating functions from Srivastava {\it et al} [Appl. Set-Valued Anal. Optim. {\bf 1} (2019), pp. 187-201.] by the method of $q$-difference equations. Moreover, we build relations between transformation formulas and homogeneous $q$-difference equations.
In this paper, we deduce the generalized $q$-difference equations for general Al-Salam--Carlitz polynomials and generalize Arjika's recently results [$q$-difference equation for homogeneous $q$-difference operators and their applications, J. Differ. Equ. Appl. {\bf 26}, 987--999 (2020)]. In addition, we obtain transformational identities by the method of $q$-difference equation. Moreover, we deduce $U(n+1)$ type generating functions and Ramanujan's integrals involving general Al-Salam--Carlitz polynomials by $q$-difference equation.
