On the SU(2) quantum group the notion of (zonal) spherical element is generalized by considering left and right invariance in the infinitesimal sense with respect to twisted primitive elements of the sl(2) quantized universal enveloping algebra. The resulting spherical elements belonging to irreducible representations of quantum SU(2) turn out to be expressible as a two-parameter family of Aske...

A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little q-Jacobi polynomials and Jacobi polynomials. Also the limits from Askey–Wilson to Wilson polynomials and from q-Racah to Racah polynomials ar...

We introduce multiple Wilson polynomials, which give a new example of multiple orthogonal polynomials (Hermite-Padé polynomials) of type II. These polynomials can be written as a Jacobi-Piñeiro transform, which is a generalization of the Jacobi transform for Wilson polynomials, found by T.H. Koornwinder. Here we need to introduce Jacobi and JacobiPiñeiro polynomials with complex parameters. Som...

The family of general Jacobi polynomials P (α,β) n where α, β ∈ C can be characterised by complex (nonhermitian) orthogonality relations (cf. [15]). The special subclass of Jacobi polynomials P (α,β) n where α, β ∈ R are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the n real zeros, have been well studied. There is another spe...

We consider a modiication of moment functionals for some classical polynomials of a discrete variable by adding a mass point at x = 0. We obtain the resulting orthogonal polynomials, identify them as hypergeometric functions and derive the second order diierence equation which these polynomials satisfy. The corresponding tridiagonal matrices and associated polynomials were also studied. x1 Intr...

Generalized nonnegative polynomials were studied in a sequence of recent papers [4], [6], [7], [8], and [9]. A number of well-known inequalities in approximation theory were extended to them, by utilizing the generalized degree in place of the ordinary one. Our motivation was to find tools to examine systems of orthogonal polynomials simultaneously, associated with generalized Jacobi, or at lea...

We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version – the Darboux transformations of Jacobi matrices – they are equivalent to Laurent polynomial modifications of the underlying measures. We address other questions which emphasize the similarities between Darboux transformations fo...

commutative properties of four-dimensional generalized symmetric pseudo-riemannian manifolds were considered. specially, in this paper, we studied skew-tsankov and jacobi-tsankov conditions in 4-dimensional pseudo-riemannian generalized symmetric manifolds.