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Register a new userRemarks on $tau$-functions for the difference Painleve equations of type $E_8$
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Authors
Masatoshi Noumi
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We investigate the structure of $tau$-functions for the elliptic difference Painleve equation of type $E_8$. Introducing the notion of ORG $tau$-functions for the $E_8$ lattice, we construct some particular solutions which are expressed in terms of elliptic hypergeometric integrals. Also, we discuss how this construction is related to the framework of lattice $tau$-functions associated with the configuration of generic nine points in the projective plane.
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Explicit determinant formulas are presented for the $tau$ functions of the generalized Painleve equations of type $A$. This result allows an interpretation of the $tau$-functions as the Plucker coordinates of the universal Grassmann manifold.
We study some properties of tau-functions of an isomonodromic deformation leading to the fifth Painleve equation. In particular, here is given an elementary proof of Miwas formula for the logarithmic differential of a tau-function.
Discrete Painleve equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according to whether the coefficient functions are linear, exponential or elliptic functions of $n$. In this paper, we focus on the elliptic type and give a review of the construction of such equations on the $E_8$ lattice. The first such construction was given by Sakai cite{SakaiH2001:MR1882403}. We focus on recent developments giving rise to more examples of elliptic discrete Painleve equations.
In an earlier paper (A. N. Kochubei, {it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirovs fractional differentiation operator $D^alpha$, $alpha >0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^alpha$ that the change of an unknown function $u=I^alpha v$ reduces the Cauchy problem for a linear equation with $D^alpha$ (for radial functions) to an integral equation whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In the present paper, we study nonlinear equations of this kind, find conditions of their local and global solvability.
The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semi-infinite matrix that models the shifts by $pm 1$ in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff-Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous Toda for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It also shown that the Kadomtesev-Petvishvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case the deformation do not satisfy a Pearson equation.
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