q-hypergeometric orthogonal

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1: 18.27 $q$-Hahn Class

§18.27 $q$-Hahn Class

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§18.27(i) Introduction

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§18.27(ii) $q$-Hahn Polynomials

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§18.27(iii) Big $q$-Jacobi Polynomials

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Discrete $q$-Hermite II

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2: 18.28 Askey–Wilson Class

… ► ►In the remainder of this section the Askey–Wilson class OP’s are defined by their $q$-hypergeometric representations, followed by their orthogonal properties. … ►

§18.28(ii) Askey–Wilson Polynomials

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§18.28(viii) $q$-Racah Polynomials

… ►Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).

3: Bibliography Z

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Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.

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External Links:

ISSN 1021-2655, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§13.9(i), §15.13.

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A proof of Andrews’ $q$-Dyson conjecture. Discrete Math. 54 (2), pp. 201–224.

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External Links:

ISSN 0012-365X, Document, MathReview (Ira Gessel), ZentralBlatt

Referenced by:

§17.14.

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Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:

Includes hypergeometric and $q$-hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).

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Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

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External Links:

ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

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On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.

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External Links:

ISSN 0021-9045, Link, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.33(iii).

Encodings:

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4: 18.33 Polynomials Orthogonal on the Unit Circle

§18.33 Polynomials Orthogonal on the Unit Circle

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§18.33(i) Definition

… ►Let $\{p_n(x)\}$ and $\{q_n(x)\}$, $n=0,1,\mathrm{\dots }$, be OP’s with weight functions $w_1(x)$ and $w_2(x)$, respectively, on $(-1,1)$. … ►For the notation, including the basic hypergeometric function ${}_2{}^{}\varphi _1^{}$, see §§17.2 and 17.4(i). … ►

Recurrence Relations

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5: Bibliography R

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The Associated Classical Orthogonal Polynomials. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 255–279.

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External Links:

MathReview (Wadim Zudilin), ZentralBlatt

Referenced by:

§18.30(viii).

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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

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Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

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Notes:

Software packages for hypergeometric and $q$-hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.

External Links:

RISC Combinatorics Group

Referenced by:

6th item.

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Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.

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External Links:

ISSN 0001-8708, Document, MathReview, ZentralBlatt

Referenced by:

§20.11(iii).

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On the zeros of the hypergeometric function. Math. Ann. 191 (1), pp. 53–58.

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External Links:

ISSN 0025-5831, Document, MathReview (R. C. Varma), ZentralBlatt

Referenced by:

§15.13.

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