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1. CJM 2010 (vol 63 pp. 200)
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)
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)
$q$-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)
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)
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 |