dorsal/arxiv
View SchemaMultivariable q-Racah polynomials
| Authors | Jan F. van Diejen, Jasper V. Stokman |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9607003 |
| URL | https://arxiv.org/abs/q-alg/9607003 |
| DOI | 10.1215/S0012-7094-98-09106-2 |
| Journal | Duke Math. J. 91 (1998), 89--136 |
Abstract
The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials is studied for parameters satisfying a truncation condition such that the orthogonality measure becomes discrete with support on a finite grid. For this parameter regime the polynomials may be seen as a multivariable counterpart of the (one-variable) $q$-Racah polynomials. We present the discrete orthogonality measure, expressions for the normalization constants converting the polynomials into an orthonormal system (in terms of the normalization constant for the unit polynomial), and we discuss the limit $q\to 1$ leading to multivariable Racah type polynomials. Of special interest is the situation that $q$ lies on the unit circle; in that case it is found that there exists a natural parameter domain for which the discrete orthogonality measure (which is complex in general) becomes real-valued and positive. We investigate the properties of a finite-dimensional discrete integral transform for functions over the grid, whose kernel is determined by the multivariable $q$-Racah polynomials with parameters in this positivity domain.
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"abstract": "The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson\npolynomials is studied for parameters satisfying a truncation condition such\nthat the orthogonality measure becomes discrete with support on a finite grid.\nFor this parameter regime the polynomials may be seen as a multivariable\ncounterpart of the (one-variable) $q$-Racah polynomials. We present the\ndiscrete orthogonality measure, expressions for the normalization constants\nconverting the polynomials into an orthonormal system (in terms of the\nnormalization constant for the unit polynomial), and we discuss the limit $q\\to\n1$ leading to multivariable Racah type polynomials. Of special interest is the\nsituation that $q$ lies on the unit circle; in that case it is found that there\nexists a natural parameter domain for which the discrete orthogonality measure\n(which is complex in general) becomes real-valued and positive. We investigate\nthe properties of a finite-dimensional discrete integral transform for\nfunctions over the grid, whose kernel is determined by the multivariable\n$q$-Racah polynomials with parameters in this positivity domain.",
"arxiv_id": "q-alg/9607003",
"authors": [
"Jan F. van Diejen",
"Jasper V. Stokman"
],
"categories": [
"q-alg",
"math.CA",
"math.QA"
],
"doi": "10.1215/S0012-7094-98-09106-2",
"journal_ref": "Duke Math. J. 91 (1998), 89--136",
"title": "Multivariable q-Racah polynomials",
"url": "https://arxiv.org/abs/q-alg/9607003"
},
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