On orthogonality relations for dual discrete \(q\)-ultraspherical polynomials (Q2495187)

The dual discrete \(q\)-ultraspherical polynomials \(D_{n}^{(s)}(\mu (x;s)| q)\) are polynomials in \(\mu (x;s)=q^{-x}+sq^{x+1}\) defined by the formula \[ D_n^{(s)}(\mu (x;s)| q)=_3\phi _2(q^{-x},sq^{x+1},q^{-n};\sqrt{s}q,-\sqrt{s}q;q,-q^{n+1}) \] in terms of the basic hypergeometric function \(_3\phi _2\). The authors derive explicit formulas that connect the dual discrete \(q\)-ultraspherical polynomials \(D_n^{(s)}(\mu (x;s)| q)\) with the \(q^{-1}\)-Hermite polynomials in the case when \(s\in \{ q^{-1},q\}\). Using these relations, all extremal orthogonality relations for these special cases of polynomials \(D_n^{(s)}(\mu (x;s)| q)\) are found.