Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution

The paper is devoted to non-Schlesinger isomonodromic deformations for resonant Fuchsian systems. There are very few explicit examples of such deformations in the literature. In this paper we construct a new example of the non-Schlesinger isomonodromic deformation for a resonant Fuchsian system of order 5 by using middle convolution for a resonant Fuchsian system of order 2. Moreover, it is known that middle convolution is an operation that preserves Schlesinger's deformation equations for non-resonant Fuchsian systems. In this paper we show that Bolibruch's non-Schlesinger deformations of resonant Fuchsian systems are, in general, not preserved by middle convolution

Recurrence coefficients of generalized Charlier polynomials and the fifth Painlev\'e equation

We investigate generalizations of the Charlier polynomials on the lattice $\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-\beta$ and on the bi-lattice $\mathbb{N}\cup (\mathbb{N}+1-\beta)$. We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to solutions of the fifth Painlev\'e equation PV (which can be transformed to the third Painlev\'e equation). Initial conditions for different lattices can be transformed to the classical solutions of PV with special values of the parameters.Comment: 14 page

Recurrence Coefficients of a New Generalization of the Meixner Polynomials

We investigate new generalizations of the Meixner polynomials on the lattice $\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-\beta$ and on the bi-lattice $\mathbb{N}\cup (\mathbb{N}+1-\beta)$. We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to the solutions of the fifth Painlev\'e equation P$_{\textup V}$. Initial conditions for different lattices can be transformed to the classical solutions of P$_{\textup V}$ with special values of the parameters. We also study one property of the B\"acklund transformation of P$_{\textup V}$

Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlev\'e VI

We investigate the recurrence coefficients of discrete orthogonal polynomials on the non-negative integers with hypergeometric weights and show that they satisfy a system of non-linear difference equations and a non-linear second order differential equation in one of the parameters of the weights. The non-linear difference equations form a pair of discrete Painlev\'e equations and the differential equation is the $\sigma$-form of the sixth Painlev\'e equation. We briefly investigate the asymptotic behavior of the recurrence coefficients as $n\to \infty$ using the discrete Painlev\'e equations

Computing recurrence coefficients of multiple orthogonal polynomials

Multiple orthogonal polynomials satisfy a number of recurrence relations, in particular there is a $(r+2)$-term recurrence relation connecting the type II multiple orthogonal polynomials near the diagonal (the so-called step-line recurrence relation) and there is a system of $r$ recurrence relations connecting the nearest neighbors (the so-called nearest neighbor recurrence relations). In this paper we deal with two problems. First we show how one can obtain the nearest neighbor recurrence coefficients (and in particular the recurrence coefficients of the orthogonal polynomials for each of the defining measures) from the step-line recurrence coefficients. Secondly we show how one can compute the step-line recurrence coefficients from the recurrence coefficients of the orthogonal polynomials of each of the measures defining the multiple orthogonality.Comment: 22 pages, 2 figures in Numerical Algorithms (2015