DLMF: Untitled Document

… ►In this section, for the function

\Gamma_{q}

see §5.18(ii). ►

17.13.1

\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-ax/c;q\right)_{\infty}\left(bx/d;q\right)_{\infty}}\,{\mathrm{d}}_{q}x=% \frac{(1-q)\left(q;q\right)_{\infty}\left(ab;q\right)_{\infty}cd\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(a;q\right)_{\infty}\left(b% ;q\right)_{\infty}(c+d)\left(-bc/d;q\right)_{\infty}\left(-ad/c;q\right)_{% \infty}},

ⓘ

Symbols:: $\int$ : integral, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, $q$ : complex base and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §17.13 and Ch.17

… ►

17.13.2

\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-xq^{\alpha}/c;q\right)_{\infty}\left(xq^{\beta}/d;q\right)_{\infty}}\,{% \mathrm{d}}_{q}x=\frac{\Gamma_{q}\left(\alpha\right)\Gamma_{q}\left(\beta% \right)}{\Gamma_{q}\left(\alpha+\beta\right)}\frac{cd}{c+d}\frac{\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(-q^{\beta}c/d;q\right)_{% \infty}\left(-q^{\alpha}d/c;q\right)_{\infty}}.

ⓘ

Symbols:: $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, $q$ : complex base and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.13 and Ch.17

… ►

17.13.3

\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}% {\left(-t;q\right)_{\infty}}\,\mathrm{d}t=\frac{\Gamma\left(\alpha\right)% \Gamma\left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-% \alpha\right)\Gamma_{q}\left(\alpha+\beta\right)},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial) and $q$ : complex base
Referenced by:: Erratum (V1.0.1) for Equation (17.13.3)
Permalink:: http://dlmf.nist.gov/17.13.E3
Encodings:: pMML, png, TeX
Errata (effective with 1.0.1):: Originally the differential was identified incorrectly as a $q$ -differential; the correct differential is $\,\mathrm{d}t$ .
See also:: Annotations for §17.13, §17.13 and Ch.17

►

17.13.4

\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-ctq^{\alpha+\beta};q\right)_{\infty}% }{\left(-ct;q\right)_{\infty}}\,{\mathrm{d}}_{q}t=\frac{\Gamma_{q}\left(\alpha% \right)\Gamma_{q}\left(\beta\right)\left(-cq^{\alpha};q\right)_{\infty}\left(-% q^{1-\alpha}/c;q\right)_{\infty}}{\Gamma_{q}\left(\alpha+\beta\right)\left(-c;% q\right)_{\infty}\left(-q/c;q\right)_{\infty}}.

ⓘ

Symbols:: $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.13.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.13, §17.13 and Ch.17

…

… ►

•

Abramowitz and Stegun (1964, Chapter 12) tabulates $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , and $I_{n}\left(x\right)-\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.1)5$ , $x^{-1}=0(.01)0.2$ to 6D or 7D.

►

•

Agrest et al. (1982) tabulates $\mathbf{H}_{n}\left(x\right)$ and $e^{-x}\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.001)5(.005)15(.01)100$ to 11D.

… ►

•

Abramowitz and Stegun (1964, Chapter 12) tabulates $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t$ and $(2/\pi)\int_{x}^{\infty}t^{-1}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.1)5$ to 5D or 7D; $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , and $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ for $x^{-1}=0(.01)0.2$ to 6D.

►

•

Agrest et al. (1982) tabulates $\int_{0}^{x}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ and $e^{-x}\int_{0}^{x}\mathbf{L}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.001)5(.005)15(.01)100$ to 11D.

… ►

•

Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function $\mathbf{H}_{n}\left(x,\alpha\right)$ for $n=0,1$ , $x=0(.2)10$ , and $\alpha=0(.2)1.4,\tfrac{1}{2}\pi$ , together with surface plots.

… ►

G. Delic (1979b) Chebyshev series for the spherical Bessel function

j_{l}(r)

. Comput. Phys. Comm. 18 (1), pp. 73–86.

ⓘ

External Links:: Document
Referenced by:: §10.76(iii).
Encodings:: BibTeX

… ►

D. Dominici, S. J. Johnston, and K. Jordaan (2013) Real zeros of

{}_{2}F_{1}

hypergeometric polynomials. J. Comput. Appl. Math. 247, pp. 152–161.

ⓘ

External Links:: ISSN 0377-0427, Document, FullText, MathReview (Nasser Saad), ZentralBlatt
Referenced by:: §15.13, §15.13.
Encodings:: BibTeX

… ►

C. F. du Toit (1993) Bessel functions

J_{n}(z)

and

Y_{n}(z)

of integer order and complex argument. Comput. Phys. Comm. 78 (1-2), pp. 181–189.

ⓘ

Notes:: Double-precision Pascal, maximum accuracy 7D.
External Links:: ISSN 0010-4655, Document, Code, MathReview (J. P. Singhal), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

T. M. Dunster (2003a) Uniform asymptotic approximations for the Whittaker functions

M_{\kappa,i\mu}(z)

and

W_{\kappa,i\mu}(z)

. Anal. Appl. (Singap.) 1 (2), pp. 199–212.

ⓘ

External Links:: ISSN 0219-5305, Document, MathReview (Hasan Taşeli), ZentralBlatt
Referenced by:: §13.20(v).
Encodings:: BibTeX

… ►

J. Dutka (1981) The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.

ⓘ

External Links:: ISSN 0003-9519, MathReview (Willard Parker), ZentralBlatt
Referenced by:: §8.17(i).
Encodings:: BibTeX

…

… ►

§17.7(i) ${{}_{2}\phi_{2}}$ Functions

►

$q$ -Analog of Bailey’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

… ►

$q$ -Analog of Gauss’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

… ►

$q$ -Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum

… ►where

m_{1},m_{2},\dots,m_{r}

are arbitrary nonnegative integers. …

… ►

•

Blanch and Rhodes (1955) includes $\mathit{Be}_{n}(t)$ , $\mathit{Bo}_{n}(t)$ , $t=\tfrac{1}{2}\sqrt{q}$ , $n=0(1)15$ ; 8D. The range of $t$ is 0 to 0.1, with step sizes ranging from 0.002 down to 0.00025. Notation: $\mathit{Be}_{n}(t)=a_{n}\left(q\right)+2q-(4n+2)\sqrt{q}$ , $\mathit{Bo}_{n}(t)=b_{n}\left(q\right)+2q-(4n-2)\sqrt{q}$ .

►

•

Ince (1932) includes eigenvalues $a_{n}$ , $b_{n}$ , and Fourier coefficients for $n=0$ or $1(1)6$ , $q=0(1)10(2)20(4)40$ ; 7D. Also $\operatorname{ce}_{n}\left(x,q\right)$ , $\operatorname{se}_{n}\left(x,q\right)$ for $q=0(1)10$ , $x=1(1)90$ , corresponding to the eigenvalues in the tables; 5D. Notation: $a_{n}=\mathit{be}_{n}-2q$ , $b_{n}=\mathit{bo}_{n}-2q$ .

… ►

•

Stratton et al. (1941) includes $b_{n}$ , $b_{n}^{\prime}$ , and the corresponding Fourier coefficients for $\mathrm{Se}_{n}(c,x)$ and $\mathrm{So}_{n}(c,x)$ for $n=0$ or $1(1)4$ , $c=0(.1~{}\textrm{or}~{}.2)4.5$ . Precision is mostly 5S. Notation: $c=2\sqrt{q}$ , $b_{n}=a_{n}+2q$ , $b^{\prime}_{n}=b_{n}+2q$ , and for $\mathrm{Se}_{n}(c,x)$ , $\mathrm{So}_{n}(c,x)$ see §28.1.

… ►

•

Ince (1932) includes the first zero for $\operatorname{ce}_{n}$ , $\operatorname{se}_{n}$ for $n=2(1)5$ or $6$ , $q=0(1)10(2)40$ ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small $q$ .

… ►For other tables prior to 1961 see Fletcher et al. (1962, §2.2) and Lebedev and Fedorova (1960, Chapter 11).

… ►

Table 18.8.1: Classical OP’s: differential equations

A(x)f^{\prime\prime}(x)+B(x)f^{\prime}(x)+C(x)f(x)+\lambda_{n}f(x)=0

.

► ►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
…
11	${\mathrm{e}}^{-n^{-1}x}x^{\ell+1}L^{(2\ell+1)}_{n-\ell-1}\left(2n^{-1}x\right)$	$1$	$0$	$\frac{2}{x}-\frac{\ell(\ell+1)}{x^{2}}$	$-\frac{1}{n^{2}}$
12	$H_{n}\left(x\right)$	$1$	$-2x$	$0$	$2n$
…
14	$\mathit{He}_{n}\left(x\right)$	$1$	$-x$	$0$	$n$

► ►Item 11 of Table 18.8.1 yields (18.39.36) for

Z=1

.

… ►

\mathfrak{S}_{n}

denotes the set of permutations of

\{1,2,\ldots,n\}

.

\sigma\in\mathfrak{S}_{n}

is a one-to-one and onto mapping from

\{1,2,\ldots,n\}

to itself. … ►An element of

\mathfrak{S}_{n}

with

a_{1}

fixed points,

a_{2}

cycles of length

2,\ldots,a_{n}

cycles of length

n

, where

n=a_{1}+2a_{2}+\cdots+na_{n}

, is said to have cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

. The number of elements of

\mathfrak{S}_{n}

with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

is given by (26.4.7). … ►A permutation with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

can be written as a product of

a_{2}+2a_{3}+\cdots+(n-1)a_{n}=n-(a_{1}+a_{2}+\cdots+a_{n})

transpositions, and no fewer. …

… ►with

b_{0}=1

and all allowable choices of

E

,

p

,

s

, and

b_{j}

. … ►Let

E_{{\rm min}}\leq E\leq E_{{\rm max}}

with

E_{{\rm min}}<0

and

E_{{\rm max}}>0

. …The integers

p

,

E_{{\rm min}}

, and

E_{{\rm max}}

are characteristics of the machine. … ►

N_{{\rm min}}\leq x\leq N_{{\rm max}}

, and …Then rounding by chopping or rounding down of

x

gives

x_{-}

, with maximum relative error

\epsilon_{M}

. …

… ►

Z. Wang and R. Wong (2002) Uniform asymptotic expansion of

J_{\nu}(\nu a)

via a difference equation. Numer. Math. 91 (1), pp. 147–193.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview (J. P. Singhal), ZentralBlatt
Referenced by:: §10.20(ii).
Encodings:: BibTeX

… ►

C. A. Wills, J. M. Blair, and P. L. Ragde (1982) Rational Chebyshev approximations for the Bessel functions

J_{0}(x)

,

J_{1}(x)

,

Y_{0}(x)

,

Y_{1}(x)

. Math. Comp. 39 (160), pp. 617–623.

ⓘ

Notes:: Tables on microfiche
External Links:: ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt
Referenced by:: 9th item.
Encodings:: BibTeX

… ►

J. A. Wilson (1991) Asymptotics for the

{}_{4}F_{3}

polynomials. J. Approx. Theory 66 (1), pp. 58–71.

ⓘ

External Links:: ISSN 0021-9045, Document, MathReview (R. G. Buschman), ZentralBlatt
Referenced by:: §18.26(v).
Encodings:: BibTeX

… ►

R. Wong and J.-M. Zhang (1994b) On the relative extrema of the Jacobi polynomials

P_{n}^{(0,-1)}(x)

. SIAM J. Math. Anal. 25 (2), pp. 776–811.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: §18.14(iii).
Encodings:: BibTeX

… ►

R. Wong (1973a) An asymptotic expansion of

{W}_{k,\,m}(z)

with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

ⓘ

External Links:: FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt
Referenced by:: §13.19.
Encodings:: BibTeX

…

… ►The choice of

x_{0}

here is critical. … ►Let

x_{0}

and

x_{1}

be such that

f_{0}=f(x_{0})

and

f_{1}=f(x_{1})

have opposite signs. …We continue with

x_{2}

and either

x_{0}

or

x_{1}

, depending which of

f_{0}

and

f_{1}

is of opposite sign to

f(x_{2})

, and so on. … ►Whether or not

f_{0}

and

f_{1}

have opposite signs,

x_{2}

is computed as in (3.8.6). … ►We construct sequences

q_{j}

and

r_{j}

,

j=n+1,n,\dots,0

, from

q_{n+1}=r_{n+1}=0

,

q_{n}=r_{n}=a_{n}

, and for

j\leq n-1

, …

.广州年世界杯看球酒吧_『wn4.com_』世界杯有女足吗_w6n2c9o_2022年11月29日7时15分46秒_p5nl57rxz_gov_hk

31—40 of 784 matching pages

31: 17.13 Integrals

32: 11.14 Tables

33: Bibliography D

34: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

§17.7(i) ${{}_{2}\phi_{2}}$ Functions

$q$ -Analog of Bailey’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Gauss’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum

35: 28.35 Tables

36: 18.8 Differential Equations

37: 26.13 Permutations: Cycle Notation

38: 3.1 Arithmetics and Error Measures

39: Bibliography W

40: 3.8 Nonlinear Equations

.广州年世界杯看球酒吧_『wn4.com_』世界杯有女足吗_w6n2c9o_2022年11月29日7时15分46秒_p5nl57rxz_gov_hk

31—40 of 784 matching pages

31: 17.13 Integrals

32: 11.14 Tables

33: Bibliography D

34: 17.7 Special Cases of Higher ϕ s r Functions

§17.7(i) ϕ 2 2 Functions

q -Analog of Bailey’s F 1 2 ⁡ ( − 1 ) Sum

q -Analog of Gauss’s F 1 2 ⁡ ( − 1 ) Sum

q -Analog of Dixon’s F 2 3 ⁡ ( 1 ) Sum

35: 28.35 Tables

36: 18.8 Differential Equations

37: 26.13 Permutations: Cycle Notation

38: 3.1 Arithmetics and Error Measures

39: Bibliography W

40: 3.8 Nonlinear Equations

34: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

§17.7(i) ${{}_{2}\phi_{2}}$ Functions

$q$ -Analog of Bailey’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Gauss’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

$q$ -Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum