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1: 17 q-Hypergeometric and Related Functions

Chapter 17 $q$-Hypergeometric and Related Functions

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2: Bibliography X

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Prolate spheroidal wavefunctions, quadrature and interpolation. Inverse Problems 17 (4), pp. 805–838.

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Notes:

Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000)

External Links:

ISSN 0266-5611, Document, MathReview (Thomas Sauer), ZentralBlatt

Referenced by:

§30.13(v).

Encodings:

BibTeX

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Variable precision computation of elementary functions. J. Numer. Methods Comput. Appl. 15 (3), pp. 161–171 (Chinese).

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Notes:

English translation in Chinese J. Numer. Math. Appl. 17 (1995), no. 1, pp. 13–24

External Links:

ISSN 1000-3266, MathReview, ZentralBlatt

Referenced by:

§4.48(iii).

Encodings:

BibTeX

3: George E. Andrews

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17 q-Hypergeometric and Related Functions

►In November 2015, Andrews was named Senior Associate Editor of the DLMF and Associate Editor for Chapter 17.

4: 19 Elliptic Integrals

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5: 19.37 Tables

… ►Tabulated for $k^2=0(.01)1$ to 6D by Byrd and Friedman (1971), to 15D for $K\left(k\right)$ and 9D for $E\left(k\right)$ by Abramowitz and Stegun (1964, Chapter 17), and to 10D by Fettis and Caslin (1964). … ►Tabulated for $\arcsin k=0(1^{\circ }){90}^{\circ }$ to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). … ►Tabulated for $k^2=0(.01)1$ to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). ►Tabulated for $\arcsin k=0(1^{\circ }){90}^{\circ }$ to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). … ►Tabulated (with different notation) for $\varphi =0({15}^{\circ }){90}^{\circ }$, ${\alpha }^2=0(.1)1$, $\arcsin k=0({15}^{\circ }){90}^{\circ }$ to 5D by Abramowitz and Stegun (1964, Chapter 17), and for $\varphi =0({15}^{\circ }){90}^{\circ }$, ${\alpha }^2=0(.1)1$, $\arcsin k=0({15}^{\circ }){90}^{\circ }$ to 7D by Zhang and Jin (1996, pp. 676–677). …

6: 19.1 Special Notation

… ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha )$, $E(\alpha )$, $\mathrm{\Pi }(n\backslash \alpha )$, $F(\varphi \backslash \alpha )$, $E(\varphi \backslash \alpha )$, and $\mathrm{\Pi }(n;\varphi \backslash \alpha )$, where $\alpha =\arcsin k$ and $n$ is the ${\alpha }^2$ (not related to $k$) in (19.1.1) and (19.1.2). …However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF. …

7: 34.14 Tables

… ►Tables of $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 17/2$ are given in Appel (1968) to 6D. …

8: Publications

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Creating Webs of Math Using LaTeX, Proceedings 6th International Congress on Industrial and Applied Mathematics, Zürich, Switzerland, July 17, 2007.

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Advances in Math Search, Proceedings 6th International Congress on Industrial and Applied Mathematics, Zürich, Switzerland, July 17, 2007.

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Methods of Relevance Ranking and Hit-content Generation in Math Search, Proceedings of Mathematical Knowledge Management (MKM2007), RISC, Hagenberg, Austria, June 27–30, 2007.

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9: 4.33 Maclaurin Series and Laurent Series

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4.33.3 $$\tanh z=z-\frac{z^3}{3}+\frac{2}{15}{z}^5-\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ .

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\tanh z$: hyperbolic tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.64

Permalink:

http://dlmf.nist.gov/4.33.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.33 and Ch.4

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10: 28.16 Asymptotic Expansions for Large $q$

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28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

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