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Mañas Baena, Manuel and Fernández Irisarri, Itsaso and González Hernández, Omar
(2021)
*Pearson equations for discrete orthogonal polynomials: I. generalized hypergeometric functions and Toda equations.*
Studies in applied mathematics
.
ISSN 0022-2526

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Official URL: http://dx.doi.org/10.1111/sapm.12471

## Abstract

The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semiinfinite matrix that models the shifts by +/- 1 in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well-known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff-Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous 1D Toda equation for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It is also shown that the Kadomtesev-Petviashvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case, the deformation does not satisfy a Pearson equation.

Item Type: | Article |
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Additional Information: | CRUE-CSIC (Acuerdos Transformativos 2021) © 2021 The Authors. |

Uncontrolled Keywords: | Recurrence coefficients; Laurent polynomials; Unit circle; Transformations; KP |

Subjects: | Sciences > Physics > Physics-Mathematical models Sciences > Physics > Mathematical physics |

ID Code: | 69373 |

Deposited On: | 14 Jan 2022 17:58 |

Last Modified: | 07 Nov 2022 16:38 |

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