Über Sturm-Liouvillesche Polynomsysteme

Bochner, S.

doi:10.1007/bf01180560

Cited by 425 publications

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“…This result can be considered as a difference equation version of Bochner's theorem [22,23,24]. In fact, the original setting of Bochner's theorem [22] is rather limited and its generalisation within the framework of ordinary quantum mechanics is given in [10], which contains various sinusoidal coordinates η(x) = x, x 2 , cos x, sinh x, e −x , etc.…”

Section: Determination Of η(X) and B(x) + D(x)mentioning

confidence: 99%

“…Determination of polynomials satisfying certain forms of difference equations for given η(x) (quadratic or q-quadratic) has a long history [23,24,27,26]. Bochner's theorem [22] on Sturm-Liouville polynomials is a precursor. The present characterisation in terms of the closure relation (4.32) is consistent with the existing ones.…”

Section: Alternative Q-krawtchoukmentioning

confidence: 99%

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Orthogonal polynomials from Hermitian matrices

Одаке

Sasaki

2008

Journal of Mathematical Physics

271

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A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schrödinger equations. The hermitian matrices (factorisable Hamiltonians) are real symmetric tri-diagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalisation measures and the normalisation constants etc. are determined explicitly.

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Section: Determination Of η(X) and B(x) + D(x)mentioning

confidence: 99%

Section: Alternative Q-krawtchoukmentioning

confidence: 99%

Orthogonal polynomials from Hermitian matrices

Одаке

Sasaki

2008

Journal of Mathematical Physics

271

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“…Bochner (1929) has shown that the classical polynomials f n (x) are completely characterised by satisfying differential equations of the form A(x)y"+B(x)y'+A n y =0, where A and B are independent of n and A n is independent of x. As the generalised Hermite polynomials do not satisfy such relation, we see that despite relations (4.1) and (4.2) we should not take them as being in any real sense equivalent to a classical system.…”

Section: Generalised Hermite Polynomials ;mentioning

confidence: 99%

Symmetric square roots of the infinite identity matrix

Pearce

Potts²

1976

J. Aust. Math. Soc.

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Some non-trivial real, symmetric square roots of the infinite identity matrix are exhibited. These may be found either from the use of involutory integral transforms and a set of real orthonormal functions or by an algebraic factorisation procedure. The two approaches are shown to be equivalent. Guinand (1956) has shown that square roots of the infinite identity matrix I can be generated from a Fourier kernel using biorthogonal functions. In particular, he gives an explicit expression for a family of asymmetric square roots of I, an example which generalises an earlier result of Barrucand (1950).In this paper we use Guinand's analytical method to develop families of real, symmetric (and hence orthogonal) square roots of I, in the sequel referred to simply as "symmetric roots". We show that all such roots can be generated by this method, and that the method is equivalent .to an algebraic factorisation procedure. Some examples and applications are also presented.A number of texts on matrices give results on square roots of finite matrices (see, for example, Perlis (1952) Theorem 9-15), and indeed there is a literature on unilateral (polynomial) equations in a finite square matrix (MacDuffee (1946) Chapter 8 gives a bibliography). Nevertheless, apart from the work of Guinand noted above, scant attention seems to have been paid to square roots (or other fractional powers) of the infinite identity matrix. In his book on infinite matrices, Cooke (1950) mentions only some trivial examples as an exercise.Square roots of I and orthogonal matrices, or rather the associated transformations, have found application in connection with summation formulae and the corresponding inversion formulae (see Guinand (1939) and Smith (1972)).

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“…A.J. Durán started the study of matrix-weights whose inner product admits a symmetric second order matrix differential operator in [Dur97], following similar considerations by S. Bochner in the scalar case in [Boc29]. A general approach to the case of higher order can be found in [DG86,GH97].…”

Section: Introductionmentioning

confidence: 99%

The Algebra of Differential Operators for a Matrix Weight: An Ultraspherical Example

Zurrián¹

2016

Int Math Res Notices

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Abstract. In this paper we study in detail algebraic properties of the algebra D(W ) of differential operators associated to a matrix weight of Gegenbauer type. We prove that two second order operators generate the algebra, indeed D(W ) is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra D(W ) is a finitely-generated torsion-free module over its center, but it is not flat and therefore it is not projective. This is the second detailed study of an algebra D(W ) and the first one coming from spherical functions and group representations. We prove that the algebras for different Gegenbauer weights and the algebras studied previously, related to Hermite weights, are isomorphic to each other. We give some general results that allow us to regard the algebra D(W ) as the centralizer of its center in the Weyl algebra. We do believe that this should hold for any irreducible weight and the case considered in this paper represents a good step in this direction.

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Über Sturm-Liouvillesche Polynomsysteme

Cited by 425 publications

References 0 publications

Orthogonal polynomials from Hermitian matrices

Orthogonal polynomials from Hermitian matrices

Symmetric square roots of the infinite identity matrix

The Algebra of Differential Operators for a Matrix Weight: An Ultraspherical Example

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