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1: 34.13 Methods of Computation

… ►Methods of computation for

\mathit{3j}

and

\mathit{6j}

symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). ►For

\mathit{9j}

symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). …

2: Bibliography H

… ►

P. I. Hadži (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

ⓘ

External Links:: MathReview (M. Lawrence Glasser)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

B. A. Hargrave and B. D. Sleeman (1977) Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

ⓘ

External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.16.
Encodings:: BibTeX

… ►

B. Hayes (2009) The higher arithmetic. American Scientist 97, pp. 364–368.

ⓘ

External Links:: FullText
Referenced by:: §3.1(iv).
Encodings:: BibTeX

… ►

L. E. Hoisington and G. Breit (1938) Calculation of Coulomb wave functions for high energies. Phys. Rev. 54 (8), pp. 627–628.

ⓘ

External Links:: Document
Referenced by:: §33.7.
Encodings:: BibTeX

… ►

K. Horata (1991) On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §24.15(iii).
Encodings:: BibTeX

…

3: 27.2 Functions

… ►

27.2.9

d\left(n\right)=\sum_{d\mathbin{|}n}1

ⓘ

Symbols:: $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $d$ : positive integer and $n$ : positive integer
A&S Ref:: 24.3.3 I.A
Permalink:: http://dlmf.nist.gov/27.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §27.2(i), §27.2 and Ch.27

… ►It is the special case

k=2

of the function

d_{k}\left(n\right)

that counts the number of ways of expressing

n

as the product of

k

factors, with the order of factors taken into account. …Note that

\sigma_{0}\left(n\right)=d\left(n\right)

. … ►Table 27.2.2 tabulates the Euler totient function

\phi\left(n\right)

, the divisor function

d\left(n\right)

(

=\sigma_{0}(n)

), and the sum of the divisors

\sigma(n)

(

=\sigma_{1}(n)

), for

n=1(1)52

. ►

Table 27.2.1: Primes.

► ►►►

$n$	$p_{n}$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
…
5	11	47	97	149	197	257	313	379	439	499
…

► …

4: Bibliography

… ►

A. Abramov (1960) Tables of

\ln\Gamma(z)

for Complex Argument. Pergamon Press, New York.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §5.22(iii).
Encodings:: BibTeX

… ►

G. B. Airy (1849) Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”. Trans. Camb. Phil. Soc. 8, pp. 595–599.

ⓘ

Referenced by:: §9.16.
Encodings:: BibTeX

… ►

F. Alhargan and S. Judah (1992) Frequency response characteristics of the multiport planar elliptic patch. IEEE Trans. Microwave Theory Tech. 40 (8), pp. 1726–1730.

ⓘ

External Links:: Document
Referenced by:: 6th item.
Encodings:: BibTeX

… ►

W. L. Anderson (1982) Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.

ⓘ

Notes:: Single-precision Fortran, maximum accuracy 8S.
External Links:: ISSN 0098-3500, Document, GAMS (module 588; package TOMS)
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

G. E. Andrews (1966b)

q

-identities of Auluck, Carlitz, and Rogers. Duke Math. J. 33 (3), pp. 575–581.

ⓘ

External Links:: Document, MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §17.9(iv).
Encodings:: BibTeX

…

5: 26.10 Integer Partitions: Other Restrictions

… ►

p\left(\mathcal{D},n\right)

denotes the number of partitions of

n

into distinct parts.

p_{m}\left(\mathcal{D},n\right)

denotes the number of partitions of

n

into at most

m

distinct parts.

p\left(\mathcal{D}k,n\right)

denotes the number of partitions of

n

into parts with difference at least

k

. … ►Note that

p\left(\mathcal{D}^{\prime}3,n\right)\leq p\left(\mathcal{D}3,n\right)

, with strict inequality for

n\geq 9

. It is known that for

k>3

p\left(\mathcal{D}k,n\right)\geq p\left(\in\!A_{1,k+3},n\right)

, with strict inequality for

n

sufficiently large, provided that

k=2^{m}-1,m=3,4,5

, or

k\geq 32

; see Yee (2004). …

6: 34.1 Special Notation

… ► ►►

$2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$	nonnegative integers.
…

►The main functions treated in this chapter are the Wigner

\mathit{3j},\mathit{6j},\mathit{9j}

symbols, respectively, … ►For other notations for

\mathit{3j}

\mathit{6j}

\mathit{9j}

symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).

7: Bibliography B

… ►

W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

ⓘ

External Links:: ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt
Referenced by:: §18.18(vii).
Encodings:: BibTeX

… ►

M. V. Berry (1975) Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces. J. Phys. A 8 (4), pp. 566–584.

ⓘ

External Links:: Document, MathReview
Referenced by:: §36.14(iii).
Encodings:: BibTeX

… ►

L. C. Biedenharn, R. L. Gluckstern, M. H. Hull, and G. Breit (1955) Coulomb functions for large charges and small velocities. Phys. Rev. (2) 97 (2), pp. 542–554.

ⓘ

External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §33.5(iv).
Encodings:: BibTeX

… ►

G. Blanch (1966) Numerical aspects of Mathieu eigenvalues. Rend. Circ. Mat. Palermo (2) 15, pp. 51–97.

ⓘ

External Links:: Document, MathReview (J. Meixner), ZentralBlatt
Referenced by:: item (e).
Encodings:: BibTeX

… ►

P. S. Bullen (1998) A Dictionary of Inequalities. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 97, Longman, Harlow.

ⓘ

External Links:: ISBN 0-8493-0634-5, MathReview (Lucio Renato Berrone), ZentralBlatt
Referenced by:: §4.18, §4.5(i), §4.5(ii).
Encodings:: BibTeX

…

8: 15.7 Continued Fractions

… ►

15.7.1

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $t_{n}$ : coefficient and $u_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/15.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

… ►

15.7.3

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $v_{n}$ : coefficient and $w_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/15.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

… ►

15.7.5

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a+1,b+1;c+1;z\right)}={x% _{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $x$ : real variable, $y$ : real variable, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

…

9: 21.9 Integrable Equations

… ►

See accompanying text — Figure 21.9.1: Two-dimensional periodic waves in a shallow water wave tank, taken from Hammack et al. (1995, p. 97) by permission of Cambridge University Press. … Magnify

…

10: Bibliography K

… ►

N. D. Kazarinoff (1988) Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.

ⓘ

External Links:: ISSN 0002-9890, FullText, MathReview (S. L. Shukla), ZentralBlatt
Referenced by:: §16.23(iii).
Encodings:: BibTeX

… ►

N. Koblitz (1993) Introduction to Elliptic Curves and Modular Forms. 2nd edition, Graduate Texts in Mathematics, Vol. 97, Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-97966-2, MathReview, ZentralBlatt
Referenced by:: §20.10(i), §20.12(i), §20.9(iii), §22.18(iv), §23.1, §23.20(ii), §23.20(v), Ch.23.
Encodings:: BibTeX

… ►

T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

ⓘ

External Links:: Document, MathReview (M. Mikolas), ZentralBlatt
Referenced by:: (18.14.8), §18.14(i), §18.17(ii), §18.18(ii).
Encodings:: BibTeX

… ►

B. G. Korenev (2002) Bessel Functions and their Applications. Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.

ⓘ

Notes:: Translated from the Russian by E. V. Pankratiev
External Links:: ISBN 0-415-28130-X, MathReview (L. Debnath), ZentralBlatt
Referenced by:: §10.73(i), §10.73(i), §10.73(i).
Encodings:: BibTeX

… ►

M. D. Kruskal and P. A. Clarkson (1992) The Painlevé-Kowalevski and poly-Painlevé tests for integrability. Stud. Appl. Math. 86 (2), pp. 87–165.

ⓘ

External Links:: ISSN 0022-2526, MathReview (Walter Strampp), ZentralBlatt
Referenced by:: §32.2(i).
Encodings:: BibTeX

…