Differential equations of infinite order for orthogonal polynomials

Krall, H. L.; Sheffer, I. M.

doi:10.1007/bf02416454

Cited by 19 publications

(14 citation statements)

References 4 publications

(4 reference statements)

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“…By using Lemma 4 we see that this equals (17). Now we will show that every solution of (61) also satisfies (60) and that every solution of (60) also satisfies (59).…”

Section: Differential Equations For Laguerre Polynomials 371mentioning

confidence: 84%

On differential equations for Sobolev-type Laguerre polynomials

Koekoek

Bavinck

1998

Trans. Amer. Math. Soc.

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Abstract. The Sobolev-type Laguerre polynomials {L α,M,N n (x)} ∞ n=0 are orthogonal with respect to the inner product,In 1990 the first and second author showed that in the case M > 0 and N = 0 the polynomials are eigenfunctions of a unique differential operator of the formwhereare independent of n. This differential operator is of order 2α + 4 if α is a nonnegative integer, and of infinite order otherwise.In this paper we construct all differential equations of the formwhere the coefficientsare independent of n and the coefficients a 0 (x), b 0 (x) and c 0 (x) are independent of x, satisfied by the Sobolev-type Laguerre polynomials {L α,M,N n (x)} ∞ n=0 . Further, we show that in the case M = 0 and N > 0 the polynomials are eigenfunctions of a linear differential operator, which is of order 2α + 8 if α is a nonnegative integer and of infinite order otherwise.Finally, we show that in the case M > 0 and N > 0 the polynomials are eigenfunctions of a linear differential operator, which is of order 4α + 10 if α is a nonnegative integer and of infinite order otherwise.

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“…By using Lemma 4 we see that this equals (17). Now we will show that every solution of (61) also satisfies (60) and that every solution of (60) also satisfies (59).…”

Section: Differential Equations For Laguerre Polynomials 371mentioning

confidence: 84%

On differential equations for Sobolev-type Laguerre polynomials

Koekoek

Bavinck

1998

Trans. Amer. Math. Soc.

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“…Equation (2.2) is identical with equation (3.1) of [1]. In Remark (i) ( [1], p. 151) it is shown that if r 2 = 0 the polynomial solutions do not form an orthogonal set.…”

Section: And In This Case X N Is Given Bymentioning

confidence: 97%

“…In Remark (i) ( [1], p. 151) it is shown that if r 2 = 0 the polynomial solutions do not form an orthogonal set. We therefore assume r 2 Φ 0.…”

Section: And In This Case X N Is Given Bymentioning

confidence: 99%

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Difference equations for some orthogonal polynomials

Krall¹,

Sheffer²

1969

Pacific J. Math.

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“…, there are infinitely many differential equations of infinite order with polynomial coefficients, To be precise, Krall and Sheffer [24] (see also [1,15] …”

Section: B(x) − Aoe(x)mentioning

confidence: 99%

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 of infinite order for orthogonal polynomials

Cited by 19 publications

References 4 publications

On differential equations for Sobolev-type Laguerre polynomials

On differential equations for Sobolev-type Laguerre polynomials

Difference equations for some orthogonal polynomials

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

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