On Orthogonality Relations for Dual Discrete q-Ultraspherical Polynomials

• Main Authors: Valentyna A. Groza, Ivan I. Kachuryk
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2006-03-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: $q$-orthogonal polynomials; dual discrete $q$-ultraspherical polynomials; $q^{-1}$-Hermite polynomials; orthogonality relation
• Online Access: http://www.emis.de/journals/SIGMA/2006/Paper034/

The dual discrete $q$-ultraspherical polynomials $D_n^{(s)}(mu (x;s)|q)$ correspond to indeterminate moment problem and, therefore, have one-parameter family of extremal orthogonality relations. It is shown that special cases of dual discrete $q$-ultraspherical polynomials $D_n^{(s)}(mu (x;s)|q)$, when $s=q^{-1}$ and $s=q$, are directly connected with $q^{-1}$-Hermite polynomials. These connections are given in an explicit form. Using these relations, all extremal orthogonality relations for these special cases of polynomials $D_n^{(s)}(mu (x;s)|q)$ are found.