# Using D-operators to construct orthogonal polynomials satisfying higher order difference or differential equations

@article{Durn2013UsingDT, title={Using D-operators to construct orthogonal polynomials satisfying higher order difference or differential equations}, author={Antonio J. Dur{\'a}n}, journal={Journal of Approximation Theory}, year={2013}, volume={174}, pages={10-53} }

## 34 Citations

### The algebras of difference operators associated to Krall–Charlier orthogonal polynomials

- MathematicsJournal of Approximation Theory
- 2018

### The higher-order differential operator for the generalized Jacobi polynomials — new representation and symmetry

- MathematicsIndagationes Mathematicae
- 2019

### On difference operators for symmetric Krall-Hahn polynomials

- MathematicsIntegral Transforms and Special Functions
- 2018

ABSTRACT The problem of finding measures whose orthogonal polynomials are also eigenfunctions of higher-order difference operators have been recently solved by multiplying the classical discrete…

### Invariance properties of Wronskian type determinants of classical and classical discrete orthogonal polynomials

- MathematicsJournal of Mathematical Analysis and Applications
- 2019

### Symmetric differential operators for Sobolev orthogonal polynomials of Laguerre- and Jacobi-type

- MathematicsIntegral Transforms and Special Functions
- 2021

ABSTRACT The Laguerre-Sobolev polynomials form an orthogonal polynomial system on the positive half-line with respect to the classical Laguerre measure with parameter and, in general, two point…

### Differential equations for discrete Jacobi–Sobolev orthogonal polynomials

- Mathematics
- 2018

The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Jacobi-Sobolev bilinear form with mass point at −1 and/or +1. In particular, we…

### Bispectrality of Charlier type polynomials

- MathematicsIntegral Transforms and Special Functions
- 2019

ABSTRACT Given a finite set of positive integers G and polynomials , , with degree of equal to g, we associate to them a sequence of Charlier type polynomials defined from the Charlier polynomials by…

### Symmetries for Casorati determinants of classical discrete orthogonal polynomials

- Mathematics
- 2013

Abstract. Given a classical discrete family (pn)n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn) and the set of numbers m + i − 1, i = 1, · · · , k and k,m ≥ 0, we consider the k ×…

### On the higher-order differential equations for the generalized Laguerre polynomials and Bessel functions

- MathematicsIntegral Transforms and Special Functions
- 2019

ABSTRACT In the enduring, fruitful research on spectral differential equations with polynomial eigenfunctions, Koornwinder's generalized Laguerre polynomials are playing a prominent role. Being…

## References

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Orthogonal polynomials satisfying fourth order differential equations were classified by H. L. Krall [K2]. They can be obtained from very special instances of the (generalized) Laguerre and the…

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We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order…

### Symmetries for Casorati determinants of classical discrete orthogonal polynomials

- Mathematics
- 2013

Abstract. Given a classical discrete family (pn)n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn) and the set of numbers m + i − 1, i = 1, · · · , k and k,m ≥ 0, we consider the k ×…

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Abstract Recently this author proved that if an OPS satisfies a differential equation of the form then, under certain conditions, an orthogonalizing weight distribution can be found that…

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- Mathematics
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### Using D-operators to construct orthogonal polynomials satisfying higher order q-difference equations

- Mathematics
- 2013

### Orthogonal Polynomials Satisfying Higher-Order Difference Equations

- Mathematics
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We introduce a large class of measures with orthogonal polynomials satisfying higher-order difference equations with coefficients independent of the degree of the polynomials. These measures are…

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The basic properties of the polynomials p n (x) that satisfy the orthogonality relations
$$ \int_a^b {{p_n}(x)} {p_m}(x)\rho (x)dx = 0\quad (m \ne n) $$
(2.0.1)
hold also for the polynomials…