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1: Bibliography H

… ►

P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

ⓘ

External Links:: MathReview (L. Berg)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

P. I. Hadži (1975b) Integrals containing the Fresnel functions

S(x)

and

C(x)

. Bul. Akad. Štiince RSS Moldoven. 1975 (3), pp. 48–60, 93 (Russian).

ⓘ

External Links:: MathReview (J. Kaczmarski)
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

… ►

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

… ►

M. H. Hull and G. Breit (1959) Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.

ⓘ

External Links:: MathReview (G. Källén)
Referenced by:: §33.2(iii), §33.23(vii), §33.23(vii), §33.24, §33.5(ii), §33.7, Ch.33.
Encodings:: BibTeX

…

2: 34.6 Definition: $\mathit{9j}$ Symbol

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

►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

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

►The

\mathit{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

3: Richard B. Paris

… ► 2022) was Reader in Mathematics at the University of Abertay Dundee, U. … ►

8 Incomplete Gamma and Related Functions

…

4: Bibliography D

… ►

S. D. Daymond (1955) The principal frequencies of vibrating systems with elliptic boundaries. Quart. J. Mech. Appl. Math. 8 (3), pp. 361–372.

ⓘ

External Links:: Document, MathReview (M. A. Hyman), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

G. Doetsch (1955) Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

ⓘ

External Links:: MathReview (I. I. Hirschman), ZentralBlatt
Referenced by:: §2.5(i).
Encodings:: BibTeX

… ►

K. Driver and K. Jordaan (2013) Inequalities for extreme zeros of some classical orthogonal and

q

-orthogonal polynomials. Math. Model. Nat. Phenom. 8 (1), pp. 48–59.

ⓘ

External Links:: ISSN 0973-5348, Document, FullText, MathReview (Ana Foulquié Moreno), ZentralBlatt
Referenced by:: §18.16(ii), §18.16(iv), §18.16(ii), §18.16(iv), §18.27(iv), §18.27(iv), §18.27(v), §18.27(v).
Encodings:: BibTeX

… ►

G. V. Dunne and K. Rao (2000) Lamé instantons. J. High Energy Phys. 2000 (1), pp. Paper 19, 8.

ⓘ

Notes:: (electronic only, article id. JHEP01(2000)019, 8 pp.).
External Links:: ISSN 1029-8479, Document, MathReview, ZentralBlatt
Referenced by:: §22.11.
Encodings:: BibTeX

… ►

T. M. Dunster (2001b) Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

ⓘ

External Links:: ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

…

5: 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 27.2.2: Functions related to division.

► ►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

►

6: Bibliography T

… ►

J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

ⓘ

Notes:: Single-precision Fortran.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 9th item.
Encodings:: BibTeX

… ►

N. M. Temme (1994a) A set of algorithms for the incomplete gamma functions. Probab. Engrg. Inform. Sci. 8, pp. 291–307.

ⓘ

Notes:: Includes Pascal program
External Links:: FullText
Referenced by:: §5.24(ii), §7.25(ii), §8.28(ii).
Encodings:: BibTeX

… ►

N. M. Temme (1995b) Bernoulli polynomials old and new: Generalizations and asymptotics. CWI Quarterly 8 (1), pp. 47–66.

ⓘ

External Links:: ISSN 0922-5366, FullText, MathReview (Michael Hauss), ZentralBlatt
Referenced by:: §24.11, §24.16(i).
Encodings:: BibTeX

… ►

J. Todd (1975) The lemniscate constants. Comm. ACM 18 (1), pp. 14–19.

ⓘ

Notes:: Collection of articles honoring Alston S. Householder. For corrigendum see same journal v. 18 (1975), no. 8, p. 462
External Links:: ISSN 0001-0782, Document, MathReview (L. Fox), ZentralBlatt
Referenced by:: §19.20(i), §19.20(iv), §19.36(iii).
Encodings:: BibTeX

… ►

O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

ⓘ

External Links:: Document
Referenced by:: §31.17(ii).
Encodings:: BibTeX

…

7: 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). …

8: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

►

D(m,n)

is the number of paths from

(0,0)

(m,n)

that are composed of directed line segments of the form

(1,0)

(0,1)

, or

(1,1)

. … ►

Table 26.6.1: Delannoy numbers

D(m,n)

► ►►

$m$	$n$
$m$	…

► … ►

26.6.4

r(n)=D(n,n)-D(n+1,n-1),

n\geq 1

ⓘ

Defines:: $r(n)$ : Schröder number (locally)
Symbols:: $n$ : nonnegative integer and $D(m,n)$ : Delannoy number
Referenced by:: §26.6(i)
Permalink:: http://dlmf.nist.gov/26.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(i), §26.6(i), §26.6 and Ch.26

… ►

26.6.10

D(m,n)=D(m,n-1)+D(m-1,n)+D(m-1,n-1),

m,n\geq 1

ⓘ

Symbols:: $m$ : nonnegative integer, $n$ : nonnegative integer and $D(m,n)$ : Delannoy number
Referenced by:: §26.6(iii)
Permalink:: http://dlmf.nist.gov/26.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(iii), §26.6 and Ch.26

…

9: 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

… ►

D. E. Amos, S. L. Daniel, and M. K. Weston (1977) Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions

I_{\nu}(x)

and

J_{\nu}(x)

x\geq 0

\nu\geq 0

. ACM Trans. Math. Software 3 (1), pp. 93–95.

ⓘ

Notes:: Single-precision Fortran, maximum accuracy 16S.
External Links:: ISSN 0098-3500, Document, Companion paper. Erratum. GAMS (module 511; package TOMS)
Referenced by:: 1st item, 1st 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

…

10: DLMF Project News

error generating summary

1—10 of 608 matching pages

1: Bibliography H

2: 34.6 Definition: 9 ⁢ j Symbol

§34.6 Definition: 9 ⁢ j Symbol

3: Richard B. Paris

4: Bibliography D

5: 27.2 Functions

6: Bibliography T

7: 26.10 Integer Partitions: Other Restrictions

8: 26.6 Other Lattice Path Numbers

Delannoy Number D ⁡ ( m , n )

9: Bibliography

10: DLMF Project News

2: 34.6 Definition: $\mathit{9j}$ Symbol

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

Delannoy Number $D(m,n)$