30 research outputs found
On classical orthogonal polynomials on bi-lattices
In [J. Phys. A: Math. Theor. 45 (2012)], while looking for spin chains that
admit perfect state transfer, Vinet and Zhedanov found an apparently new
sequence of orthogonal polynomials, that they called para-Krawtchouk
polynomials, defined on a bilinear lattice. In this note we present necessary
and sufficient conditions for the regularity of solutions of the corresponding
functional equation. Moreover, the functional Rodrigues formula and a closed
formula for the recurrence coefficients are presented. As a consequence, we
characterize all solutions of the functional equation, including as very
particular cases the Meixner, Charlier, Krawtchouk, Hahn, and para-Krawtchouk
polynomials.Comment: arXiv admin note: substantial text overlap with arXiv:2102.0003
ON A REGULARISATION OF A NONLINEAR DIFFERENTIAL EQUATION RELATED TO THE NON-HOMOGENEOUS AIRY EQUATION
In this paper we study a nonlinear differential equation related to a non-homogeneous Airy equation. The linear equation has two families of solutions. We apply a procedure of resolution of points of indeterminacy to a system of first order differential equations equivalent to the nonlinear equation and study how the corresponding families of solutions are transformed
On differential systems related to generalized Meixner and deformed Laguerre orthogonal polynomials
In this paper we present a connection between systems of differential equations for the recurrence coefficients of polynomials orthogonal with respect to the generalized Meixner and the deformed Laguerre weights. It is well-known that the recurrence coefficients of both generalized Meixner and deformed Laguerre orthogonal polynomials can be expressed in terms of solutions of the fifth Painlevé equation but no explicit relation between systems of differential equations for the recurrence coefficients was known. We also present certain limits in which the recurrence coefficients can be expressed in terms of solutions of the Painlevé XXXIV equation, which in the deformed Laguerre case extends previous studies and in the generalized Meixner case is a new result
Hamiltonian structure for a differential system from a modified Laguerre weight via the geometry of the modified third Painlevé equation
Recurrence coefficients of semi-classical orthogonal polynomials are often related to the solutions of special nonlinear second-order differential equations known as the Painlevé equations. Each Painlevé equation can be written in a standard form as a non-autonomous Hamiltonian system, so it is natural to ask whether differential systems satisfied by the recurrence coefficients also possess Hamiltonian structures. We consider recurrence coefficients for a modified Laguerre weight which satisfy a differential system known to be related to the modified third Painlevé equation and identify a Hamiltonian structure for it by constructing its space of initial conditions. We also discuss a transformation from this system to the modified third Painlevé equation which simultaneously identifies a discrete system for the recurrence coefficients with a discrete Painlevé equation
On Certain Wronskians of Multiple Orthogonal Polynomials
We consider determinants of Wronskian type whose entries are multiple orthogonal polynomials associated with a path connecting two multi-indices. By assuming that the weight functions form an algebraic Chebyshev (AT) system, we show that the polynomials represented by the Wronskians keep a constant sign in some cases, while in some other cases oscillatory behavior appears, which generalizes classical results for orthogonal polynomials due to Karlin and Szegő. There are two applications of our results. The first application arises from the observation that the m-th moment of the average characteristic polynomials for multiple orthogonal polynomial ensembles can be expressed as a Wronskian of the type II multiple orthogonal polynomials. Hence, it is straightforward to obtain the distinct behavior of the moments for odd and even m in a special multiple orthogonal ensemble - the AT ensemble. As the second application, we derive some Turán type inequalities for multiple Hermite and multiple Laguerre polynomials (of two kinds). Finally, we study numerically the geometric configuration of zeros for the Wronskians of these multiple orthogonal polynomials. We observe that the zeros have regular configurations in the complex plane, which might be of independent interest
The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlev\'e 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 Painlev\'e 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 B\"acklund transformation of the fourth
Painlev\'e equation.Comment: 18 page
On the Linearization of the Painleve' III-VI Equations and Reductions of the Three-Wave Resonant System
We extend similarity reductions of the coupled (2+1)-dimensional three-wave
resonant interaction system to its Lax pair. Thus we obtain new 3x3 matrix
Fuchs--Garnier pairs for the third and fifth Painleve' equations, together with
the previously known Fuchs--Garnier pair for the fourth and sixth Painleve'
equations. These Fuchs--Garnier pairs have an important feature: they are
linear with respect to the spectral parameter. Therefore we can apply the
Laplace transform to study these pairs. In this way we found reductions of all
pairs to the standard 2x2 matrix Fuchs--Garnier pairs obtained by M. Jimbo and
T. Miwa. As an application of the 3x3 matrix pairs, we found an integral
auto-transformation for the standard Fuchs--Garnier pair for the fifth
Painleve' equation. It generates an Okamoto-like B\"acklund transformation for
the fifth Painleve' equation. Another application is an integral transformation
relating two different 2x2 matrix Fuchs--Garnier pairs for the third Painleve'
equation.Comment: Typos are corrected, journal and DOI references are adde
Movable algebraic singularities of second-order ordinary differential equations
Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a
(generally branched) solution with leading order behaviour proportional to
(z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic
at z_0 and a_N(z_0)\ne 0. We consider the subclass of equations for which each
possible leading order term of this form corresponds to a one-parameter family
of solutions represented near z_0 by a Laurent series in fractional powers of
z-z_0. For this class of 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 work generalizes previous results of S.
Shimomura. The only other possible kind of movable singularity that might occur
is an accumulation point of algebraic singularities that can be reached by
analytic continuation along infinitely long paths ending at a finite point in
the complex plane. This behaviour cannot occur for constant coefficient
equations in the class considered. However, an example of R. A. Smith shows
that such singularities do occur in solutions of a simple autonomous
second-order differential equation outside the class we consider here
Ladder operators and differential equations for multiple orthogonal polynomials
In this paper, we obtain the ladder operators and associated compatibility
conditions for the type I and the type 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.Comment: 28 page