Pearson equations for discrete orthogonal polynomials: I. Generalized hypergeometric functions and Toda equations
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 matr...
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Published in | Studies in applied mathematics (Cambridge) Vol. 148; no. 3; pp. 1141 - 1179 |
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Language | English |
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01.04.2022
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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. |
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AbstractList | 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. 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. 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. |
Author | Mañas, Manuel González‐Hernández, Omar F. Fernández‐Irisarri, Itsaso |
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Cites_doi | 10.1007/s13398-022-01296-4 10.1016/j.jmaa.2014.10.087 10.1016/S0375-9601(97)00341-1 10.1016/j.aim.2016.06.029 10.1016/j.aim.2014.06.019 10.1111/sapm.12202 10.1142/S1664360719500073 10.1016/S0377-0427(02)00597-6 10.1007/s00365-011-9145-8 10.1016/0377-0427(93)E0247-J 10.1007/s002200050738 10.1155/S1073792801000460 10.1063/1.533175 10.1016/j.jat.2017.10.007 10.1088/0266-5611/6/4/008 10.1017/9780511979156 10.1016/j.aim.2013.02.020 10.1090/S0002-9939-2012-11468-6 10.1016/0021-9045(86)90088-2 10.1016/j.aim.2011.03.008 10.1080/10236198.2018.1441836 10.1007/s002200050609 10.2140/pjm.2014.268.389 10.1142/S2010326320400031 10.1017/CBO9781107337411 10.1007/BFb0076565 10.1007/978-3-642-74748-9 10.1088/1751-8121/aab9ca |
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Snippet | The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete... Abstract The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical... |
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SubjectTerms | 3D Nijhoff–Capel discrete Toda equation Cholesky factorization contigous relations Deformation discrete orthogonal polynomials generalized hypergeometric functions Hypergeometric functions Independent variables Mathematical analysis Norms Pearson equations Polynomials Toda hierarchy |
Title | Pearson equations for discrete orthogonal polynomials: I. Generalized hypergeometric functions and Toda equations |
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