On a differential equation for Koornwinder’s generalized Laguerre polynomials

Koekoek, J.; Koekoek, Roelof

doi:10.1090/s0002-9939-1991-1047003-9

Cited by 46 publications

(53 citation statements)

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“…Later many contributions in this direction were made by T. Koornwinder [28], L. Littlejohn [30], J. Koekoek, R. Koekoek [27], etc. In recent times there is much activity in generalizations and versions of the classical result of Bochner, see, e.g., [17,18,20,24,25].…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

Horozov¹

2016

SIGMA

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Abstract. We construct new families of vector orthogonal polynomials that have the property to be eigenfunctions of some differential operator. They are extensions of the Hermite and Laguerre polynomial systems. A third family, whose first member has been found by Y. Ben Cheikh and K. Douak is also constructed. The ideas behind our approach lie in the studies of bispectral operators. We exploit automorphisms of associative algebras which transform elementary vector orthogonal polynomial systems which are eigenfunctions of a differential operator into other systems of this type.

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Section: Introductionmentioning

confidence: 99%

Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

Horozov¹

2016

SIGMA

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“…In [5] J. Koekoek and R. Koekoek showed that the polynomials {L%'M(x)}™=0 satisfy a unique differential equation of the form…”

Section: Introductionmentioning

confidence: 99%

Differential equations for symmetric generalized ultraspherical polynomials

Koekoek¹

1994

Trans. Amer. Math. Soc.

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Abstract. We look for differential equations satisfied by the generalized Jacobi polynomials {/*« (jc)}J10 which are orthogonal on the interval [-1, 1] with respect to the weight function 2^n,Vi^+i)(I-xr(1+<)#+1#*(*+l)+W(,-'1)' where a > -1, ß > -\, Af > 0, and N>0.In the special case that ß = a and N = M we find all differential equations of the formwhere the coefficients {cfa)}^ are independent of the degree n . We show that if M > 0 only for nonnegative integer values of a there exists exactly one differential equation which is of finite order 2a + 4 .By using quadratic transformations we also obtain differential equations for the polynomials {PZ,±l/2'°'N(x)}%LQ for all a>-l and N>0.

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“…So, the example analyzed in this paper (semiclassical Jacobi-type OP with respect to a moment functional of class s = 4) is, probably, a good starting point, since other semiclassical cases like Laguerre-type, Hermite-type (see Álvarez-Nodarse and Marcellán, 1995;Koekoek and Koekoek, 1991;Koekoek, 1988), and, of course, the very classical OP should be connected by limit transitions to the semiclassical Jacobi-type OP as the starting family. The present paper helps us to understand the analysis of semiclassical OP (very close to the classical ones), and then one has a good basis for studying other extensions, for which there are quite a few possibilities for research.…”

Section: Conclusion and Open Problemsmentioning

confidence: 92%

“…Also the particular cases of the Krall-type polynomials (A. M. Krall, 1981; H. L. Krall, 1940) have been obtained from this general case as special cases or limit cases. The Laguerre case was considered in detail in Koekoek and Koekoek (1991), Koekoek (1988Koekoek ( , 1990.…”

Section: Introductionmentioning

confidence: 99%

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Arvesú

Marcellán

Álvarez-Nodarse

2002

Acta Applicandae Mathematicae

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Abstract. The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional Uwhere J α,β is the Jacobi linear functional, i.e.and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (−1, 1) (inner asymptotics) and C \ [−1, 1] (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional U is a generalization of one studied by T. H. Koornwinder when A 2 = B 2 = 0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi-Markov function by a rational function with two double poles at ±1. The denominators of the [n − 1/n] Padé approximants are our orthogonal polynomials. (2000): 33C45, 42C05. Mathematics Subject Classifications

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On a differential equation for Koornwinder’s generalized Laguerre polynomials

Abstract: Abstract.Koornwinder's generalized Laguerre polynomials {L°' (x)}^=0 are orthogonal on the interval [0, oo) with respect to the weight function r,'+|.xae~j: + Nô(x), a > -1 , N > 0. We show that these polynomials for N > 0 satisfy a unique differential equation of the form

Cited by 46 publications

References 7 publications

Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

Differential equations for symmetric generalized ultraspherical polynomials

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