S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

ⓘ

External Links:: ISSN 0024-1318, MathReview (James M. Horner)
Referenced by:: §13.9(i).
Encodings:: BibTeX

… ►

D. E. Amos (1974) Computation of modified Bessel functions and their ratios. Math. Comp. 28 (125), pp. 239–251.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (L. Lorch), ZentralBlatt
Referenced by:: §10.74(v).
Encodings:: BibTeX

… ►

V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

ⓘ

Notes:: Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.
External Links:: Document, MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

… ►

R. Askey (1980) Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11 (6), pp. 938–951.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt
Referenced by:: §17.13, §17.13.
Encodings:: BibTeX

…

13: 26.12 Plane Partitions

… ►

Table 26.12.1: Plane partitions.

► ►►►►

$n$	$\operatorname{pp}\left(n\right)$	$n$	$\operatorname{pp}\left(n\right)$	$n$	$\operatorname{pp}\left(n\right)$
…
11	859	28	24 83234	45	17740 79109
…
13	2485	30	56 68963	47	36379 93036
…

► … ►

26.12.26

\operatorname{pp}\left(n\right)\sim\frac{\left(\zeta\left(3\right)\right)^{7/3% 6}}{2^{11/36}(3\pi)^{1/2}n^{25/36}}\exp\left(3\left(\zeta\left(3\right)\right)% ^{1/3}\left(\tfrac{1}{2}n\right)^{2/3}+\zeta'\left(-1\right)\right),

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\sim$ : asymptotic equality, $\exp\NVar{z}$ : exponential function, $\operatorname{pp}\left(\NVar{n}\right)$ : number of plane partitions of $n$ , $n$ : nonnegative integer and $\pi$ : plane partition
Referenced by:: §26.12(iv), Erratum (V1.0.5) for Equation (26.12.26)
Permalink:: http://dlmf.nist.gov/26.12.E26
Encodings:: pMML, png, TeX
Errata (effective with 1.0.5):: The original statement of this equation, $\operatorname{pp}\left(n\right)\sim\left(\frac{\zeta\left(3\right)}{2^{11}n^{2% 5}}\right)^{1/36}\*\exp\left(3\left(\frac{\zeta\left(3\right)n^{2}}{4}\right)^% {1/3}+\zeta'\left(-1\right)\right)$ , has been corrected.
Reported 2011-09-05 by Suresh Govindarajan
See also:: Annotations for §26.12(iv), §26.12 and Ch.26

…

14: 34.6 Definition: $\mathit{9j}$ Symbol

… ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

…

15: Bibliography O

… ►

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

… ►

A. B. Olde Daalhuis (2004a) Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.

ⓘ

External Links:: ISSN 0013-0915, Document, MathReview (Niro Yanagihara), ZentralBlatt
Referenced by:: §2.9(i).
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

… ►

M. Onoe (1956) Modified quotients of cylinder functions. Math. Tables Aids Comput. 10, pp. 27–28.

ⓘ

External Links:: ISSN 0891-6837, MathReview Entry, ZentralBlatt
Referenced by:: §10.29(i), §10.6(i).
Encodings:: BibTeX

… ►

H. Oser (1960) Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.

ⓘ

Notes:: Includes Algol code, minimum accuracy 9S. For certification see same journal 13(1970), p. 448.
External Links:: ISSN 0001-0782, Document
Referenced by:: §10.77(ii).
Encodings:: BibTeX

…

16: 26.6 Other Lattice Path Numbers

… ►

Table 26.6.1: Delannoy numbers

D(m,n)

► ►►►►

$m$	$n$
$m$	…
1	1	3	5	7	9	11	13	15	17	19	21
…
5	1	11	61	231	681	1683	3653	7183	13073	22363	36365
…

► … ►

Table 26.6.2: Motzkin numbers

M(n)

► ►►►

$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$
…
3	4	7	127	11	5798	15	3 10572	19	181 99284

► … ►

Table 26.6.3: Narayana numbers

N(n,k)

► ►►►

$n$	$k$
$n$	…
8	0	1	28	196	490	490	196	28	1
…

► … ►

Table 26.6.4: Schröder numbers

r(n)

► ►►►

$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$
…
$3$	$22$	$7$	$8558$	$11$	$52\;93446$	$15$	$39376\;03038$	$19$	$323\;67243\;17174$

► …

17: Bibliography J

… ►

D. S. Jones (1964) The Theory of Electromagnetism. International Series of Monographs on Pure and Applied Mathematics, Vol. 47. A Pergamon Press Book, The Macmillan Co., New York.

ⓘ

External Links:: MathReview (W. E. Williams), ZentralBlatt
Referenced by:: §23.21(iii).
Encodings:: BibTeX

… ►

W. B. Jones and W. J. Thron (1974) Numerical stability in evaluating continued fractions. Math. Comp. 28 (127), pp. 795–810.

ⓘ

External Links:: FullText, MathReview (N. N. Abdelmalek), ZentralBlatt
Referenced by:: §3.10(iii).
Encodings:: BibTeX

… ►

W. B. Jones and W. J. Thron (1980) Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.

ⓘ

External Links:: ISBN 0-201-13510-8, MathReview (E. Frank), ZentralBlatt
Referenced by:: §1.12(ii), §1.12(iii), §1.12(iv), §1.12(v), §13.17, §13.17, §13.5, §13.5, §4.25, §5.10.
Encodings:: BibTeX

… ►

G. Julia (1918) Memoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8 (1), pp. 47–245 (French).

ⓘ

External Links:: FullText, ZentralBlatt
Referenced by:: §3.8(viii).
Encodings:: BibTeX

18: Bibliography K

… ►

G. A. Kalugin, D. J. Jeffrey, and R. M. Corless (2012) Bernstein, Pick, Poisson and related integral expressions for Lambert

W

. Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

ⓘ

External Links:: ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt
Referenced by:: §4.13.
Encodings:: BibTeX

… ►

E. L. Kaplan (1948) Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

ⓘ

External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §19.12.
Encodings:: BibTeX

… ►

R. P. Kerr (1963) Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), pp. 237–238.

ⓘ

External Links:: Document, MathReview (C. Gilbert), ZentralBlatt
Referenced by:: §31.17(ii).
Encodings:: BibTeX

… ►

K. S. Kölbig (1968) Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.

ⓘ

Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §25.21(v).
Encodings:: BibTeX

… ►

T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

ⓘ

External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §15.14, §15.14.
Encodings:: BibTeX

…

19: Bibliography C

… ►

L. Carlitz (1960) Note on Nörlund’s polynomial

B_{n}^{(z)}

. Proc. Amer. Math. Soc. 11 (3), pp. 452–455.

ⓘ

External Links:: FullText, MathReview (M. P. Drazin), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

… ►

P. A. Clarkson (2003b) The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

… ►

J. A. Cochran (1963) Further formulas for calculating approximate values of the zeros of certain combinations of Bessel functions. IEEE Trans. Microwave Theory Tech. 11 (6), pp. 546–547.

ⓘ

External Links:: ISSN 0018-9480, FullText
Referenced by:: §10.21(x).
Encodings:: BibTeX

… ►

M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

… ►

F. Cooper, A. Khare, and A. Saxena (2006) Exact elliptic compactons in generalized Korteweg-de Vries equations. Complexity 11 (6), pp. 30–34.

ⓘ

External Links:: ISSN 1076-2787, Document, MathReview
Referenced by:: §19.6(i).
Encodings:: BibTeX

…

20: Bibliography B

… ►

R. Barakat (1961) Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

ⓘ

External Links:: FullText, MathReview (J. C. P. Miller), ZentralBlatt
Referenced by:: §8.7.
Encodings:: BibTeX

… ►

B. C. Berndt, S. Bhargava, and F. G. Garvan (1995) Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt
Referenced by:: §15.8(v), §20.11(iii).
Encodings:: BibTeX

… ►

F. Bethuel (1998) Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

ⓘ

External Links:: ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §17.17.
Encodings:: BibTeX

… ►

A. Bhattacharyya and L. Shafai (1988) Theoretical and experimental investigation of the elliptical annual ring antenna. IEEE Trans. Antennas and Propagation 36 (11), pp. 1526–1530.

ⓘ

External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

… ►

R. L. Bishop (1981) Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

ⓘ

External Links:: ISSN 0035-9149, Document, MathReview
Referenced by:: Sidebar 9.SB1.
Encodings:: BibTeX

…

.阿根廷世界杯预选赛_『网址:68707.vip』世界杯塞尔维亚对哥_b5p6v3_2022年11月28日21时21分47秒_6smwawga4

11—20 of 138 matching pages

11: 26.3 Lattice Paths: Binomial Coefficients

12: Bibliography

13: 26.12 Plane Partitions

14: 34.6 Definition: 9 ⁢ j Symbol

15: Bibliography O

16: 26.6 Other Lattice Path Numbers

17: Bibliography J

18: Bibliography K

19: Bibliography C

20: Bibliography B

14: 34.6 Definition: $\mathit{9j}$ Symbol