representation as q-hypergeometric functions

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1: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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§17.6(v) Integral Representations

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2: 18.28 Askey–Wilson Class

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

3: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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Sum Related to (17.6.4)

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$q$-Pfaff–Saalschütz Sum

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Nonterminating Form of the $q$-Saalschütz Sum

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Continued Fractions

►For continued-fraction representations of a ratio of ${}_3{}^{}\varphi _2^{}$ functions, see Cuyt et al. (2008, pp. 399–400). …

4: 18.27 $q$-Hahn Class

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18.27.13 $$p_n(x)=p_n(x;a,b;q)={}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},ab{q}^{n+1}}{aq};q,qx)={(-b)}^{-n}{q}^{-n(n+1)/2}\frac{{(qb;q)}_n}{{(qa;q)}_n}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},ab{q}^{n+1},qbx}{qb,0};q,q).$$

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Defines:

$p_n(x;a,b;q)$: little $q$-Jacobi polynomial

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.28.27), (18.28.28), Erratum (V1.2.0) for Equation (18.27.13)

Permalink:

http://dlmf.nist.gov/18.27.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

The ${}_3{}^{}\varphi _2^{}$ representation was added.

See also:

Annotations for §18.27(iv), §18.27 and Ch.18

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5: 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.

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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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).

Encodings:

BibTeX

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Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.

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Referenced by:

§35.7(ii).

Encodings:

BibTeX

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Tablitsy vyrozhdennoi gipergeometricheskoi funktsii. Vyčisl. Centr Akad. Nauk SSSR, Moscow (Russian).

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Notes:

Translated title: Tables of the confluent hypergeometric function

External Links:

MathReview (A. H. Stroud), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

6: Bibliography R

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

Encodings:

BibTeX

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

BibTeX

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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