Author
Galina Filipuk
Other affiliations: Polish Academy of Sciences, University of Kent, Loughborough University ...read more
Bio: Galina Filipuk is an academic researcher from University of Warsaw. The author has contributed to research in topics: Orthogonal polynomials & Classical orthogonal polynomials. The author has an hindex of 15, co-authored 92 publications receiving 665 citations. Previous affiliations of Galina Filipuk include Polish Academy of Sciences & University of Kent.
Papers published on a yearly basis
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TL;DR: The authors showed 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.
Abstract: 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.
60 citations
TL;DR: In this paper, it was shown that the Backlund transformations of the sixth Painleve equation can also be obtained from this viewpoint, and that they preserve the deformation equations.
Abstract: Middle convolution and addition are operations for Fuchsian systems of differential equations which preserve the number of accessory parameters. In this paper we show that they also preserve the deformation equations. Several Backlund transformations of the sixth Painleve equation are obtained from this viewpoint.
47 citations
TL;DR: In this article, 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.
Abstract: 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.
42 citations
TL;DR: In this paper, the authors present a list of hypergeometric-to-Heun pull-back transformations with a free continuous parameter, and illustrate most of them by a Heun-tohypergeometric reduction formula.
Abstract: 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.
41 citations
TL;DR: In this article, two families of rational solutions and associated special polynomials for the equations in the symmetric fourth Painleve hierarchy are studied, and the structure of the roots of these polynomial is shown to be highly regular in the complex plane.
Abstract: In this paper two families of rational solutions and associated special polynomials for the equations in the symmetric fourth Painleve hierarchy are studied. The structure of the roots of these polynomials is shown to be highly regular in the complex plane. Further representations are given of the associated special polynomials in terms of Schur functions. The properties of these polynomials are compared and contrasted with the special polynomials associated with rational solutions of the fourth Painleve equation.
40 citations
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2,618 citations
01 Dec 1982
1,915 citations
Book•
01 Jan 1966
TL;DR: Boundary value problems in physics and engineering were studied in this article, where Chorlton et al. considered boundary value problems with respect to physics, engineering, and computer vision.
Abstract: Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s
733 citations
01 Jan 2011
TL;DR: A discussion of the most important formulae and construction of plots for some functions of mathematical physics relevant to quantum mechanics can be found in this paper, where the eigenfunctions of the one-dimensional harmonic oscillator and the radial eigenfunction of the harmonics oscillator in three dimensions are discussed.
Abstract: Discussion of the most important formulae and construction of plots for some functions of mathematical physics relevant to quantum mechanics. These functions are Hermite polynomials, Legendre polynomials and Legendre functions, spherical harmonics, Bessel functions and spherical Bessel functions and Laguerre polynomials. Directly related to some of these and also discussed are the eigenfunctions of the one-dimensional harmonic oscillator and the radial eigenfunctions of the harmonics oscillator in three dimensions and of the hydrogen atom.
276 citations
266 citations