relation to q-hypergeometric function

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1: 18.27 $q$-Hahn Class

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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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§18.27(iv) Little $q$-Jacobi Polynomials

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§18.27(v) $q$-Laguerre Polynomials

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Discrete $q$-Hermite II

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2: 18.28 Askey–Wilson Class

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§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: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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Ramanujan’s ${}_1{}^{}\psi _1^{}$ Summation

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Quintuple Product Identity

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Bailey’s Bilateral Summations

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Sum Related to (17.6.4)

… ►For similar formulas see Verma and Jain (1983).

4: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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Sum Related to (17.6.4)

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5: George E. Andrews

… ►An expert on $q$-series, he is the author of $q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. … ►Andrews was elected to the American Academy of Arts and Sciences in 1997, and to the National Academy of Sciences (USA) in 2003. …Andrews served as President of the AMS from February 1, 2009 to January 31, 2011, and became a Fellow of the AMS in 2012. … ►

17 q-Hypergeometric and Related Functions

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6: 17.4 Basic Hypergeometric Functions

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17.4.2 $$\underset{q\to 1-}{lim}{}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{q^{a_0},q^{a_1},\mathrm{\dots },q^{a_r}}{q^{b_1},\mathrm{\dots },q^{b_s}};q,{(q-1)}^{s-r}z)={}_{r+1}{}^{}F_s^{}(\genfrac{}{}{0pt}{}{a_0,a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};z).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, $q$: complex base, $r$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Source:

Koekoek et al. (2010, p. 15)

Referenced by:

Erratum (V1.1.11) for Equation (17.4.2)

Permalink:

http://dlmf.nist.gov/17.4.E2

Encodings:

pMML, png, TeX

Generalization (effective with 1.1.11):

This limit relation which was previously accurate for ${}_{r+1}{}^{}\varphi _r^{}$ was updated to be accurate for the more general ${}_{r+1}{}^{}\varphi _s^{}$.

See also:

Annotations for §17.4(i), §17.4 and Ch.17

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7: 17.13 Integrals

§17.13 Integrals

►In this section, for the function ${\mathrm{\Gamma }}_q$ see §5.18(ii). …or, when , … ►

Ramanujan’s Integrals

… ►Askey (1980) conjectured extensions of the foregoing integrals that are closely related to Macdonald (1982). …

8: Bibliography Z

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On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

BibTeX

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The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.

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External Links:

MathReview (J. Browkin), ZentralBlatt

Referenced by:

§25.12(i).

Encodings:

BibTeX

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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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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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9: 18.33 Polynomials Orthogonal on the Unit Circle

… ►For an alternative and more detailed approach to the recurrence relations, see §18.33(vi). … ►

Szegő–Askey

… ►For the hypergeometric function ${}_2{}^{}F_1^{}$ see §§15.1 and 15.2(i). … ►For the notation, including the basic hypergeometric function ${}_2{}^{}\varphi _1^{}$, see §§17.2 and 17.4(i). … ►Equivalent to the recurrence relations (18.33.23), (18.33.24) are the inverse Szegő recurrence relations …

10: Bibliography R

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Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications. Contemporary Mathematics, Vol. 138, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-5159-4, MathReview, ZentralBlatt

Referenced by:

Ch.35, Profile Donald St. P. Richards.

Encodings:

BibTeX

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Total positivity properties of generalized hypergeometric functions of matrix argument. J. Statist. Phys. 116 (1-4), pp. 907–922.

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External Links:

ISSN 0022-4715, Document, MathReview (Stamatis Koumandos), ZentralBlatt

Referenced by:

§35.9.

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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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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