Orthogonal Basic Hypergeometric Laurent Polynomials
The Askey-Wilson polynomials are orthogonal polynomials in$x = cos heta$, which are given as a terminating $_4phi_3$ basic hypergeometric series. The non-symmetric Askey-Wilson polynomials are Laurent polynomials in $z=e^{iheta}$, which are given as a sum of two terminating $_4phi_3$'s. They sa...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2012-12-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.3842/SIGMA.2012.092 |
| Summary: | The Askey-Wilson polynomials are orthogonal polynomials in$x = cos heta$, which are given as a terminating $_4phi_3$ basic hypergeometric series. The non-symmetric Askey-Wilson polynomials are Laurent polynomials in $z=e^{iheta}$, which are given as a sum of two terminating $_4phi_3$'s. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single $_4phi_3$'s which are Laurent polynomials in~$z$ are given, which imply the non-symmetric Askey-Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study of the results for non-symmetricAskey-Wilson polynomials which were previously obtained by affine Hecke algebra techniques. |
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| ISSN: | 1815-0659 |