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Search: MSC category 33D45 ( Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) )

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1. CJM 2010 (vol 63 pp. 200)

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

Keywords:6-$j$ and 9-$j$ symbols, $q$-analogues, balanced and very-well-poised basic hypergeometric series, orthonormal polynomials in one and two variables, Racah and $q$-Racah polynomials and their extensions
Categories:33D45, 33D50

2. CJM 2010 (vol 63 pp. 181)

Ismail, Mourad E. H.; Obermaier, Josef
Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials
We characterize the continuous $q$-ultraspherical polynomials in terms of the special form of the coefficients in the expansion $\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$ 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.

Keywords:continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficients
Categories:33D45, 42C05

3. CJM 2002 (vol 54 pp. 709)

Ismail, Mourad E. H.; Stanton, Dennis
$q$-Integral and Moment Representations for $q$-Orthogonal Polynomials
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 $\mathcal{E}_q$.

Keywords:$q$-integral, $q$-orthogonal polynomials, associated polynomials, $q$-difference equations, generating functions, Al-Salam-Chihara polynomials, continuous $q$-ultraspherical polynomials
Categories:33D45, 33D20, 33C45, 30E05

4. CJM 1997 (vol 49 pp. 520)

Ismail, Mourad E. H.; Stanton, Dennis
Classical orthogonal polynomials as moments
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.

Keywords:Classical orthogonal polynomials, \ACP, continuous, $q$-ultraspherical polynomials, generating functions, multilinear, generating functions, transformation formulas, umbral calculus
Categories:33D45, 33D20, 33C45, 30E05

5. CJM 1997 (vol 49 pp. 373)

Stokman, Jasper V.; Koornwinder, Tom H.
Limit transitions for BC type multivariable orthogonal 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.

Categories:33D45, 33C50

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