DLMF: Untitled Document

… ►A transformation of a convergent sequence

\{s_{n}\}

with limit

\sigma

into a sequence

\{t_{n}\}

is called limit-preserving if

\{t_{n}\}

converges to the same limit

\sigma

. … ►This transformation is accelerating if

\{s_{n}\}

is a linearly convergent sequence, i. … ►Then the transformation of the sequence

\{s_{n}\}

into a sequence

\{t_{n,2k}\}

is given by … ►Then

t_{n,2k}=\varepsilon_{2k}^{(n)}

. … ►We give a special form of Levin’s transformation in which the sequence

s=\{s_{n}\}

of partial sums

s_{n}=\sum_{j=0}^{n}a_{j}

is transformed into: …

… ►

K. Okamoto (1986) Studies on the Painlevé equations. III. Second and fourth Painlevé equations,

P_{{\rm II}}

and

P_{\mathrm{IV}}

. Math. Ann. 275 (2), pp. 221–255.

ⓘ

External Links:: ISSN 0025-5831, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(ii), §32.10(iv), §32.6(iii), §32.6(v), §32.7(viii).
Encodings:: BibTeX

►

K. Okamoto (1987a) Studies on the Painlevé equations. I. Sixth Painlevé equation

P_{{\rm VI}}

. Ann. Mat. Pura Appl. (4) 146, pp. 337–381.

ⓘ

External Links:: ISSN 0003-4622, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(vi), §32.6(v), §32.7(vii), §32.7(viii).
Encodings:: BibTeX

►

K. Okamoto (1987b) Studies on the Painlevé equations. II. Fifth Painlevé equation

P_{\rm V}

. Japan. J. Math. (N.S.) 13 (1), pp. 47–76.

ⓘ

External Links:: ISSN 0289-2316, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(v), §32.6(v), §32.7(viii).
Encodings:: BibTeX

►

K. Okamoto (1987c) Studies on the Painlevé equations. IV. Third Painlevé equation

P_{{\rm III}}

. Funkcial. Ekvac. 30 (2-3), pp. 305–332.

ⓘ

External Links:: ISSN 0532-8721, FullText, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(iii), §32.2(iii), §32.6(iv), §32.7(viii).
Encodings:: BibTeX

… ►

S. Olver (2011) Numerical solution of Riemann-Hilbert problems: Painlevé II. Found. Comput. Math. 11 (2), pp. 153–179.

ⓘ

External Links:: ISSN 1615-3375, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §32.17, §32.17.
Encodings:: BibTeX

…

… ►

S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.

ⓘ

External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
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

…

… ►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.

… ►

L. J. Landau (1999) Ratios of Bessel functions and roots of

\alpha J_{\nu}(x)+xJ^{\prime}_{\nu}(x)=0

. J. Math. Anal. Appl. 240 (1), pp. 174–204.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §10.14, §10.15, §10.21(xii).
Encodings:: BibTeX

… ►

J. LeCaine (1945) A table of integrals involving the functions

E_{n}(x)

.

ⓘ

Notes:: National Research Council of Canada, Division of Atomic Energy, Document no. MT-131 (NRC 1553)
Referenced by:: §8.19(x).
Encodings:: BibTeX

… ►

A. Leitner and J. Meixner (1960) Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.

ⓘ

External Links:: ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §30.12.
Encodings:: BibTeX

… ►

M. Lerch (1887) Note sur la fonction

\mathfrak{K}(w,x,s)=\sum_{k=0}^{\infty}\frac{e^{2k\pi ix}}{(w+k)^{s}}

. Acta Math. 11 (1-4), pp. 19–24 (French).

ⓘ

External Links:: ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt
Referenced by:: §25.14(i), Notations K.
Encodings:: BibTeX

… ►

N. A. Lukaševič (1967b) On the theory of Painlevé’s third equation. Differ. Uravn. 3 (11), pp. 1913–1923 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 3(11), pp. 994–999
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.10(iii), §32.8(iii), §32.9(i), §32.9(ii).
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. …

… ►The

B_{k}^{n}

are the differentiated Lagrangian interpolation coefficients: … ►where

\xi_{0}

and

\xi_{1}\in I

. ►For the values of

n_{0}

and

n_{1}

used in the formulas below … ►For partial derivatives we use the notation

u_{t,s}=u(x_{0}+th,y_{0}+sh)

. …

… ►

•

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).

… ►

W. Gautschi (1966) Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

ⓘ

External Links:: ISSN 0001-0782, Document
Referenced by:: §33.26(ii).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2002c) Computing complex Airy functions by numerical quadrature. Numer. Algorithms 30 (1), pp. 11–23.

ⓘ

External Links:: ISSN 1017-1398, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §9.17(iii).
Encodings:: BibTeX

… ►

D. Goss (1978) Von Staudt for

{\mathbf{F}}_{q}[{T}]

. Duke Math. J. 45 (4), pp. 885–910.

ⓘ

External Links:: ISSN 0012-7094, Document, MathReview (Neal Koblitz), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

ⓘ

External Links:: FullText, MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

… ►

V. I. Gromak (1975) Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 11(11), pp. 285–287
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.7(iii), §32.7(vi).
Encodings:: BibTeX

…

.世界杯竞猜投注分析_『wn4.com_』中央1台世界杯时间表_w6n2c9o_2022年11月29日6时6分30秒_qsuoiwyqw_gov_hk

31—40 of 788 matching pages

31: 3.9 Acceleration of Convergence

32: Bibliography O

33: Bibliography D

34: 17.13 Integrals

35: 11.14 Tables

36: Bibliography L

37: 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

38: 3.4 Differentiation

39: 28.35 Tables

40: Bibliography G

.世界杯竞猜投注分析_『wn4.com_』中央1台世界杯时间表_w6n2c9o_2022年11月29日6时6分30秒_qsuoiwyqw_gov_hk

31—40 of 788 matching pages

31: 3.9 Acceleration of Convergence

32: Bibliography O

33: Bibliography D

34: 17.13 Integrals

35: 11.14 Tables

36: Bibliography L

37: 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

38: 3.4 Differentiation

39: 28.35 Tables

40: Bibliography G

37: 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