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21: 28.2 Definitions and Basic Properties

§28.2 Definitions and Basic Properties

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§28.2(ii) Basic Solutions $w_{\text{I}}$, $w_{\text{II}}$

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22: 13.2 Definitions and Basic Properties

§13.2 Definitions and Basic Properties

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23: 13.14 Definitions and Basic Properties

§13.14 Definitions and Basic Properties

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24: 28.20 Definitions and Basic Properties

§28.20 Definitions and Basic Properties

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25: 12.16 Mathematical Applications

… ►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). … ►Sleeman (1968b) considers certain orthogonality properties of the PCFs and corresponding eigenvalues. …

26: Bibliography S

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Very high accuracy Chebyshev expansions for the basic trigonometric functions. Math. Comp. 34 (149), pp. 237–244.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (Claude Carasso), ZentralBlatt

Referenced by:

§4.47(i).

Encodings:

BibTeX

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Spline Functions: Basic Theory. John Wiley & Sons Inc., New York.

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Notes:

Pure and Applied Mathematics, A Wiley-Interscience Publication. Reprinted by Robert E. Krieger Publishing Co. Inc., 1993.

External Links:

ISBN 0-471-76475-2, MathReview (D. D. Stancu), ZentralBlatt

Referenced by:

§24.17(ii), §3.11(vi).

Encodings:

BibTeX

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Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.

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External Links:

ISSN 0012-7094, Link, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

BibTeX

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Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

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External Links:

ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt

Referenced by:

§24.10(ii).

Encodings:

BibTeX

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An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§17.3(i).

Encodings:

BibTeX

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27: 18.27 $q$-Hahn Class

… ►All these systems of OP’s have orthogonality properties of the form …Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property. … ►They are defined by their $q$-hypergeometric representations, followed by their orthogonality properties. … ►

18.27.3 $$Q_n(x)=Q_n(x;\alpha ,\beta ,N;q)={}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},\alpha \beta q^{n+1},x}{\alpha q,q^{-N}};q,q),$$ $n=0,1,\mathrm{\dots },N$.

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Defines:

$Q_n(x;\alpha ,\beta ,N;q)$: $q$-Hahn polynomial

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.27(ii)

Permalink:

http://dlmf.nist.gov/18.27.E3

Encodings:

pMML, png, TeX

See also:

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

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18.27.14_5 $$p_n(x;a,0;q)={}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},0}{aq};q,qx).$$

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Symbols:

$p_n(x;a,b;q)$: little $q$-Jacobi polynomial, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.12.12), (14.20.1))

Referenced by:

§18.27(iv)

Permalink:

http://dlmf.nist.gov/18.27.E14_5

Encodings:

pMML, png, TeX

See also:

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

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28: 16.4 Argument Unity

… ►The basic transformation is given by … ►The characterizing properties (18.22.2), (18.22.10), (18.22.19), (18.22.20), and (18.26.14) of the Hahn and Wilson class polynomials are examples of the contiguous relations mentioned in the previous three paragraphs. …

29: 18.38 Mathematical Applications

… ►The monic Chebyshev polynomial $2^{1-n}{T}_n\left(x\right)$, $n\ge 1$, enjoys the ‘minimax’ property on the interval $[-1,1]$, that is, $|2^{1-n}{T}_n\left(x\right)|$ has the least maximum value among all monic polynomials of degree $n$. … ►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. …

30: 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. For further properties see Koekoek et al. (2010, Chapter 14). … ►

18.28.1_5 $$R_n(z)=R_n(z;a,b,c,d|q)=\frac{p_n(\frac{1}{2}(z+z^{-1});a,b,c,d|q)}{a^{-n}{(ab,ac,ad;q)}_n}={}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},abcd{q}^{n-1},az,a{z}^{-1}}{ab,ac,ad};q,q).$$

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Symbols:

$p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $z$: complex variable, $q$: real variable and $n$: nonnegative integer

Source:

Koornwinder and Mazzocco (2018, (1))

Referenced by:

(18.28.26), §18.28(ii), §18.28(ix), §18.28(x), §18.38(iii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.28.E1_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.28(ii), §18.28 and Ch.18

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18.28.9 $$Q_n(\frac{1}{2}(a{q}^{-y}+a^{-1}{q}^y);a,b|q^{-1})={(-1)}^n{b}^n{q}^{-\frac{1}{2}n(n-1)}{({(ab)}^{-1};q)}_n{}_3{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},q^{-y},a^{-2}{q}^y}{{(ab)}^{-1}};q,q^{n}a{b}^{-1}).$$

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Defines:

$Q_n(x;a,b|q^{-1})$: $q^{-1}$-Al-Salam–Chihara 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, $y$: real variable, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(iv), §18.28 and Ch.18

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