Galina Filipuk

The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation

We show that the coefficients of the three-term recurrence relation for orthogonal polynomials with respect to a semi-classical extension of the Laguerre weight satisfy the fourth Painleve equation when viewed as functions of one of the parameters in the weight. We compare different approaches to derive this result, namely, the ladder operators approach, the isomonodromy deformations approach and combining the Toda system for the recurrence coefficients with a discrete equation. We also discuss a relation between the recurrence coefficients for the Freud weight and the semi-classical Laguerre weight and show how it arises from the Backlund transformation of the fourth Painleve equation.

A classification of coverings yielding Heun-to-hypergeometric reductions

Pull-back transformations between Heun and Gauss hypergeometric equations give useful expressions of Heun functions in terms of better understood hypergeometric functions. This article classifies, up to Mobius automorphisms, the coverings P1-to-P1 that yield pull-back transformations from hypergeometric to Heun equations with at least one free parameter (excluding the cases when the involved hypergeometric equation has cyclic or dihedral monodromy). In all, 61 parametric hypergeometric-to-Heun transformations are found, of maximal degree 12. Among them, 28 pull-backs are compositions of smaller degree transformations between hypergeometric and Heun functions. The 61 transformations are realized by 48 different Belyi coverings (though 2 coverings should be counted twice as their moduli field is quadratic). The same Belyi coverings appear in several other contexts. For example, 38 of the coverings appear in Herfutners list of elliptic surfaces over P1 with four singular fibers, as their j-invariants. In passing, we demonstrate an elegant way to show that there are no coverings P1-to-P1 with some branching patterns.

Parametric Transformations between the Heun and Gauss Hypergeometric Functions

The hypergeometric and Heun functions are classical special functions. Transformation formulas between them are commonly induced by pull-back transformations of their differential equations, with respect to some coverings P1-to-P1. This gives expressions of Heun functions in terms of better understood hypergeometric functions. This article presents the list of hypergeometric-to-Heun pull-back transformations with a free continuous parameter, and illustrates most of them by a Heun-to-hypergeometric reduction formula. In total, 61 parametric transformations exist, of maximal degree 12.

Recurrence coefficients of generalized Meixner polynomials and Painlevé equations

We consider a semi-classical version of the Meixner weight depending on two parameters and the associated set of orthogonal polynomials. These polynomials satisfy a three-term recurrence relation. We show that the coefficients appearing in this relation satisfy a discrete Painleve equation, which is a limiting case of an asymmetric dPIV equation. Moreover, when viewed as functions of one of the parameters, they satisfy one of Chazys second-degree Painleve equations, which can be reduced to the fifth Painleve equation PV.

Recurrence coefficients of a new generalization of the Meixner polynomials

We investigate new generalizations of the Meixner polynomials on the lattice N, on the shifted lattice N + 1 and on the bi-lattice N( (N + 1 ). 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 V. Initial conditions for different lattices can be transformed to the classical solutions of PV with special values of the parameters. We also study one property of the Backlund transformation of PV. we study a new generalization of the Meixner weight. The recurrence coefficients of the corre- sponding orthogonal polynomials can be viewed as functions of one of the parameters. We show that the recurrence coefficients are related to solutions of the fifth Painlev e equation. Another generalization of the Meixner weight is presented in (2). The paper is organized as follows. In the introduction we shall first review orthogonal poly- nomials for the generalized Meixner weight on different lattices and their main properties fol- lowing (12). Next we shall briefly recall the fifth Painlev e equation and its Backlund transfor- mation. Further, by using the Toda system, we show that the recurrence coefficients can be expressed in terms of solutions of the fifth Painlev e equation. Finally we study initial conditions of the recurrence coefficients for different lattices and describe one property of the Backlund transformation of PV.

Movable algebraic singularities of second-order ordinary differential equations

Any nonlinear equation of the form y″=∑n=0Nan(z)yn has a solution with leading behavior proportional to (z−z0)−2/(N−1) about a point z0, where the coefficients an are analytic at z0 and aN(z0)≠0. Equations are considered for which each possible leading term of this form extends to a Laurent series solution in fractional powers of z−z0. For these equations we show that the only movable singularities that can be reached by analytic continuation along finite-length curves are of the algebraic type just described. This generalizes results of Shimomura [“On second order nonlinear differential equations with the quasi-Painleve property II,” RIMS Kokyuroku 1424, 177 (2005)]. The possibility that these algebraic singularities could accumulate along infinitely long paths ending at a finite point is considered. Smith [“On the singularities in the complex plane of the solutions of y″+y′f(y)+g(y)=P(x),” Proc. Lond. Math. Soc. 3, 498 (1953)] showed that such singularities do occur in solutions of a simple equation out...

Factorization of (q, h)-difference operators – an algebraic approach

In this paper we introduce -deformation of Weyl algebra and study the ladders in this algebra, which give the factorization of certain q- and h-difference operators of second order.

Ladder operators and differential equations for multiple orthogonal polynomials

In this paper, we obtain the ladder operators and associated compatibility conditions for types I and II multiple orthogonal polynomials. These ladder equations extend known results for orthogonal polynomials and can be used to derive the differential equations satisfied by multiple orthogonal polynomials. Our approach is based on Riemann–Hilbert problems and the Christoffel–Darboux formula for multiple orthogonal polynomials, and the nearest-neighbor recurrence relations. As an illustration, we give several explicit examples involving multiple Hermite and Laguerre polynomials, and multiple orthogonal polynomials with exponential weights and cubic potentials.

The generalized Krawtchouk polynomials and the fifth Painleve equation

We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth Painlevé equation when viewed as functions of one of the parameters in the weight.

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.