Differential equations having orthogonal polynomial solutions

Kwon, Kil Hyun; Lee, D. W.; Littlejohn, Lance L.

doi:10.1016/s0377-0427(96)00096-9

Cited by 14 publications

(2 citation statements)

References 19 publications

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“…Since the 1990's, a significant amount of bibliography (e.g. [9]) allowed us to understand the orthogonal polynomial eigenfunctions of ordinary differential equations. With regard to the notion of d-orthogonal polynomials which generalises the standard orthogonality and it is defined by means of d functionals, for any positive integer d, much is still unknown.…”

Section: Introductionmentioning

confidence: 99%

Symbolic Approach to 2-Orthogonal Polynomial Solutions of a Third Order Differential Equation

Mesquita

2022

Math.Comput.Sci.

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In a recent work, a generic differential operator on the vectorial space of polynomial functions was presented and applied in the study of differential relations fulfilled by polynomial sequences either orthogonal or 2-orthogonal. Considering a third order differential operator that does not increase the degree of polynomials, we search for polynomial eigenfunctions with the help of symbolic computations, assuming that those polynomials constitute a 2-orthogonal polynomial sequence. Two examples are extensively described.

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Section: Introductionmentioning

confidence: 99%

Symbolic Approach to 2-Orthogonal Polynomial Solutions of a Third Order Differential Equation

Mesquita

2022

Math.Comput.Sci.

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“…In particular, the monomiality principle combines the use of two operators written in terms of the derivative operator and its inverse so that these two operators play similar roles to those of differentiation and multiplication by x on monomials, respectively. In other important contributions like [9], Kwon et al consider a generic differential operator L N [y] = N i=1 a i (x)y (i) (x) and discuss the existence of orthogonal polynomial sequences {P n } n≥0 , with deg (P n (x)) = n, such that L N [P n ](x) = λ n P n , for each n. It is then worth notice that in view of the actual state of art, with regard to differential operators L such that deg (L (P n )) = n, operating into an orthogonal sequence, we have already some acute results, as for instance, the nonexistence of orthogonal solutions among such differential equations of odd order [9]. More recently, some incursions on the study of an Appell-type behavior of orthogonal polynomials [6,20] allowed to gain new insights concerning differential relations fulfilled by orthogonal polynomial sequences.…”

Section: Introductionmentioning

confidence: 99%

Around operators not increasing the degree of polynomials

Mesquita,

Maroni

2016

Preprint

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We present a generic operator J simply defined as a linear map not increasing the degree from the vectorial space of polynomial functions into itself and we address the problem of finding the polynomial sequences that coincide with the (normalized) J-image of themselves. The technique developed assembles different types of operators and initiates with a transposition of the problem to the dual space. It is also provided examples where the results are applied to the case where J's expansion is limited to three terms.

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Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

Kwon¹,

Littlejohn²,

Yoon³

2001

Journal of Approximation Theory

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Let y=s+n be a point mass perturbation of a classical moment functional s by a distribution n with finite support. We find necessary conditions for the polynomials {Q n (x)} . n=0 , orthogonal relative to y, to be a Bochner-Krall orthogonal polynomial system (BKOPS); that is, {Q n (x)} . n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients:In particular, when n is of order 0 as a distribution, we find necessary and sufficient conditions for {Q n (x)} . n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.

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Differential equations having orthogonal polynomial solutions

Cited by 14 publications

References 19 publications

Symbolic Approach to 2-Orthogonal Polynomial Solutions of a Third Order Differential Equation

Symbolic Approach to 2-Orthogonal Polynomial Solutions of a Third Order Differential Equation

Around operators not increasing the degree of polynomials

Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

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