Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy

A $\mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided-difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the $\mathbb{D}$-semi-classical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the $\mathbb{D}$-semi-classical class it is entirely natural to define a generalisation of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first non-trivial deformation of the Askey-Wilson orthogonal polynomial system defined by the $q$-quadratic divided-difference operator, the Askey-Wilson operator, and derive the coupled first order divided-difference equations characterising its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the $E^{(1)}_7$ $q$-Painlev\'e system.Comment: Submitted to Duke Mathematical Journal on 5th April 201

Semi-classical Laguerre polynomials and a third order discrete integrable equation

A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials, leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main example we use is a semi-classical Laguerre weight to derive a third order difference equation with a corresponding Lax pair.Comment: 11 page

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

Bi-orthogonal systems on the unit circle, Regular Semi-Classical Weights and Integrable Systems - II

We derive the Christoffel-Geronimus-Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities $\{z_1, ..., z_M \}$ the bi-orthogonal system is known to be isomonodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel-Geronimus-Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system - the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota-Miwa equations are derived for the $\tau$-functions or equivalently Toeplitz determinants of the system.Comment: to appear J. Approx. Theor

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$_{\textup V}$