We characterize the continuous q-ultraspherical polynomials in terms of the special form of the coefficients in the expansion ${{\mathcal{D}}_{q}}{{P}_{n}}\left( x \right)$ in the basis $\left\{ {{P}_{n}}\left( x \right) \right\},{{\mathcal{D}}_{q}}$ being the Askey-Wilson divided difference operator. The polynomials are assumed to be symmetric, and the connection coefficients are multiples of the reciprocal of the square of the ${{L}^{2}}$ norm of the polynomials. A similar characterization is given for the discrete $q$-ultraspherical polynomials. A new proof of the evaluation of the connection coefficients for big $q$-Jacobi polynomials is given.

Using standard transformation and summation formulas for basic hypergeometric series we obtain an explicit polynomial form of the $q$-analogue of the $\text{9-}\,j$ symbols, introduced by the author in a recent publication. We also consider a limiting case in which the $\text{9-}\,j$ symbol factors into two Hahn polynomials. The same factorization occurs in another limit case of the corresponding $q$-analogue.

We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for $q$-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the $q$-exponential function $\text{ }{{\varepsilon }_{q}}$.

We continue to study the simplest closure conditions for chains of spectral transformations of the Laurent biorthogonal polynomials $\left( \text{LBP} \right)$. It is shown that the 1-1-periodic $q$-closure condition leads to the $\text{LBP}$ introduced by Pastro. We introduce classes of semi-classical and Laguerre-Hahn $\text{LBP}$ associated to generic closure conditions of the chain of spectral transformations.

We show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous q-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures.We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.

Limit transitions will be derived between the five parameter family of Askey-Wilson polynomials, the four parameter family of big q-Jacobi polynomials and the three parameter family of little q-Jacobi polynomials in n variables associated with root system BC. These limit transitions generalize the known hierarchy structure between these families in the one variable case. Furthermore it will be proved that these three families are q-analogues of the three parameter family of BC type Jacobi polynomials in n variables. The limit transitions will be derived by taking limits of q-difference operators which have these polynomials as eigenfunctions.

Nonnegative product linearization of the Askey-Wilson polynomials is shown for a wide range of parameters. As a corollary we obtain Rahman's result on the continuous q-Jacobi polynomials with α ≥ β > — 1 and α + β + 1 ≥ 0.

We use generating functions to express orthogonality relations in the form of q-beta. integrals. The integrand of such a q-beta. integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegö and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.

From Koornwinder's interpretation of big q-Legendre polynomials as spherical elements on the quantum SU(2) group an addition formula is derived for the big g-Legendre polynomial. The formula involves Al-Salam-Carlitz polynomials, little q-Jacobi polynomials and dual q-Krawtchouk polynomials. For the little q-ultraspherical polynomials a product formula in terms of a big q-Legendre polynomial follows by q-integration. The addition and product formula for the Legendre polynomials are obtained when q tends to 1.

In this paper a q-analogue of Gegenbauer's addition formula for Bessel functions is obtained by using the orthogonality relation for the q-Ultraspherical polynomials of Rogers'. Also some product formulas and an integral representation for the Hahn-Exton q-Bessel functions are obtained.