Orthogonal Basic Hypergeometric Laurent Polynomials

• Main Authors: Mourad E.H. Ismail, Dennis Stanton
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2012-12-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: Askey-Wilson polynomials; orthogonality
• Online Access: http://dx.doi.org/10.3842/SIGMA.2012.092

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.