New orthogonality relations for the continuous and the discrete \(q\)-ultraspherical polynomials (Q982606)

The authors distinguish two kinds of \(q\)-ultraspherical polynomials. The first family is the well-known family of continuous \(q\)-ultraspherical polynomials introduced and studied by Rogers. In the cases where the parameters do not satisfy the conditions of Favard's theorem, the authors prove an orthogonality relation in terms of the Askey-Wilson divided difference operator \(D_q\). This is called a \(D_q\)-Sobolev orthogonality relation. The second family is a special case of the big \(q\)-Jacobi polynomials, which can be seen as a discrete \(q\)-analogue of the ultraspherical or Gegenbauer polynomials. Also in this case the authors prove a \(D_q\)-Sobolev orthogonality relation in the case that the parameters do not satisfy the conditions of Favard's theorem.