DLMF: Untitled Document

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Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols

The Legendre polynomial $P_{n}$ was mistakenly identified as the associated Legendre function $P_{n}$ in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

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Section 34.1

The relation between Clebsch-Gordan and $\mathit{3j}$ symbols was clarified, and the sign of $m_{3}$ was changed for readability. The reference Condon and Shortley (1935) for the Clebsch-Gordan coefficients was replaced by Edmonds (1974) and Rotenberg et al. (1959) and the references for $\mathit{3j}$ , $\mathit{6j}$ , $\mathit{9j}$ symbols were made more precise in §34.1.

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Section 34.1

The reference for Clebsch-Gordan coefficients, Condon and Shortley (1935), was replaced by Edmonds (1974) and Rotenberg et al. (1959). The references for $\mathit{3j}$ , $\mathit{6j}$ , $\mathit{9j}$ symbols were made more precise.

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Equation (34.7.4)

34.7.4

\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{m_{r1},m_{r2},r=1,2,3}\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}

Originally the third $\mathit{3j}$ symbol in the summation was written incorrectly as $\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}.$

Reported 2015-01-19 by Yan-Rui Liu.

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Equation (34.3.7)

34.3.7

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ j_{1}&-j_{1}-m_{3}&m_{3}\end{pmatrix}=(-1)^{j_{1}-j_{2}-m_{3}}\left(\frac{(2j_% {1})!(-j_{1}+j_{2}+j_{3})!(j_{1}+j_{2}+m_{3})!(j_{3}-m_{3})!}{(j_{1}+j_{2}+j_{% 3}+1)!(j_{1}-j_{2}+j_{3})!(j_{1}+j_{2}-j_{3})!(-j_{1}+j_{2}-m_{3})!(j_{3}+m_{3% })!}\right)^{\frac{1}{2}}

In the original equation the prefactor of the above 3j symbol read $(-1)^{-j_{2}+j_{3}+m_{3}}$ . It is now replaced by its correct value $(-1)^{j_{1}-j_{2}-m_{3}}$ .

Reported 2014-06-12 by James Zibin.

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J. L. Fields and Y. L. Luke (1963a) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.

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External Links:: Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.11(iii).
Encodings:: BibTeX

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J. L. Fields and Y. L. Luke (1963b) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.

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External Links:: Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.11(iii).
Encodings:: BibTeX

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J. L. Fields (1965) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III. J. Math. Anal. Appl. 12 (3), pp. 593–601.

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External Links:: Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §16.11(iii).
Encodings:: BibTeX

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J. P. M. Flude (1998) The Edmonds asymptotic formulas for the

3j

and

6j

symbols. J. Math. Phys. 39 (7), pp. 3906–3915.

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External Links:: ISSN 0022-2488, Document, MathReview (A. J. Coleman), ZentralBlatt
Referenced by:: §34.8.
Encodings:: BibTeX

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T. Fort (1948) Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

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External Links:: MathReview (W. J. Trjitzinsky), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

…

… ►A domain

D

, say, is an open set in

\mathbb{C}

that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. … ►When

\sum a_{n}z^{n}

and

\sum b_{n}z^{n}

both converge … ►If a double series is absolutely convergent, then it is also convergent and its sum is given by either of the repeated sums … ►for any finite contour

C

in

D

. … ►Let

(a,b)

be a finite or infinite interval, and

f_{0}(t),f_{1}(t),\dots

be real or complex continuous functions,

t\in(a,b)

. …

… ►

R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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J. K. G. Watson (1999) Asymptotic approximations for certain

6

-

j

and

9

-

j

symbols. J. Phys. A 32 (39), pp. 6901–6902.

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External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §34.8, §34.8.
Encodings:: BibTeX

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J. V. Wehausen and E. V. Laitone (1960) Surface Waves. In Handbuch der Physik, Vol. 9, Part 3, pp. 446–778.

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External Links:: Online edition, MathReview (F. Ursell), ZentralBlatt
Referenced by:: §11.12.
Encodings:: BibTeX

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E. J. Weniger (1996) Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

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External Links:: Document
Referenced by:: §2.11(vi), §2.11(vi).
Encodings:: BibTeX

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J. A. Wilson (1991) Asymptotics for the

{}_{4}F_{3}

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

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

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… ►Assume that we wish to integrate (3.7.1) along a finite path

\mathscr{P}

from

z=a

to

z=b

in a domain

D

. … ►Then for

s=2,3,\dots

, … ►Write

\tau_{j}=z_{j+1}-z_{j}

,

j=0,1,\dots,P

, expand

w(z)

and

w^{\prime}(z)

in Taylor series (§1.10(i)) centered at

z=z_{j}

, and apply (3.7.2). … ►This is a set of

2P

equations for the

2P+2

unknowns,

w(z_{j})

and

w^{\prime}(z_{j})

,

j=0,1,\dots,P

. … ►Let

(a,b)

be a finite or infinite interval and

q(x)

be a real-valued continuous (or piecewise continuous) function on the closure of

(a,b)

. …

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$3j$ and $6j$ Symbols

►The

3j

symbol (34.2.6), with an alternative expression as a terminating

{{}_{3}F_{2}}

of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. The orthogonality relations in §34.3(iv) for the

3j

symbols can be rewritten in terms of orthogonality relations for Hahn or dual Hahn polynomials as given by §§18.2(i), 18.2(iii) and Table 18.19.1 or by §18.25(iii), respectively. … … ► …

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See accompanying text — Figure 25.11.1: Hurwitz zeta function $\zeta\left(x,a\right)$ , $a$ = 0. 3, 0. … Magnify

… ►where

h, k

are integers with

1\leq h\leq k

and

n=1,2,3,\dots

. … ►

25.11.36Removed because it is just (25.15.1) combined with (25.15.3).

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Symbols:: $L\left(\NVar{s},\NVar{\chi}\right)$ : Dirichlet $L$ -function and $k$ : nonnegative integer
Referenced by:: §25.11(x), Erratum (V1.0.23) for Equation (25.11.36), Erratum (V1.0.23) for Equation (25.11.36)
Permalink:: http://dlmf.nist.gov/25.11.E36
Removal (effective with 1.1.4):: This equation has been removed because it is just (25.15.1) combined with (25.15.3).
Clarification (effective with 1.0.23):: The Dirichlet $L$ -function was inserted on the left-hand side. The upper-index of the finitesum which originally was $k$ , was replaced with $k-1$ since $\chi(k)=0$ .
See also:: Annotations for §25.11(x), §25.11 and Ch.25

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§25.11(xi) Sums

… ►For further sums see Prudnikov et al. (1990, pp. 396–397) and Hansen (1975, pp. 358–360). …

… ►Under certain conditions on their parameters the orthogonality range for the Wilson polynomials and continuous dual Hahn polynomials is

(0,\infty)\cup S

, where

S

is a specific finite set, e. … ►

18.25.5

h_{n}=\frac{n!\,2\pi\prod_{j<\ell}\Gamma\left(n+a_{j}+a_{\ell}\right)}{(2n-1+% \sum_{j}a_{j})\Gamma\left(n-1+\sum_{j}a_{j}\right)}.

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\ell$ : nonnegative integer, $n$ : nonnegative integer and $h_{n}$
Referenced by:: §18.25(ii)
Permalink:: http://dlmf.nist.gov/18.25.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

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18.25.9

\sum_{y=0}^{N}p_{n}(y(y+\gamma+\delta+1))p_{m}(y(y+\gamma+\delta+1))\*\frac{% \gamma+\delta+1+2y}{\gamma+\delta+1+y}\omega_{y}=h_{n}\delta_{n,m}.

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $y$ : real variable, $N$ : positive integer, $\delta$ : arbitrary small positive constant, $p_{n}(x)$ : polynomial of degree $n$ , $m$ : nonnegative integer, $n$ : nonnegative integer and $h_{n}$
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/18.25.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §18.25(iii), §18.25 and Ch.18

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18.25.11

\omega_{y}=\frac{{\left(\alpha+1\right)_{y}}{\left(\beta+\delta+1\right)_{y}}{% \left(\gamma+1\right)_{y}}{\left(\gamma+\delta+2\right)_{y}}}{{\left(-\alpha+% \gamma+\delta+1\right)_{y}}{\left(-\beta+\gamma+1\right)_{y}}{\left(\delta+1% \right)_{y}}y!},

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $y$ : real variable and $\delta$ : arbitrary small positive constant
Referenced by:: §18.25(i)
Permalink:: http://dlmf.nist.gov/18.25.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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18.25.15

h_{n}=\frac{n!\,(N-n)!\,{\left(\gamma+\delta+2\right)_{N}}}{N!\,{\left(\gamma+% 1\right)_{n}}{\left(\delta+1\right)_{N-n}}}.

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $N$ : positive integer, $\delta$ : arbitrary small positive constant, $n$ : nonnegative integer and $h_{n}$
Referenced by:: §18.25(iii)
Permalink:: http://dlmf.nist.gov/18.25.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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§24.4(iii) Sums of Powers

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24.4.11

\sum_{\begin{subarray}{c}k=1\\ \left(k,m\right)=1\end{subarray}}^{m}k^{n}=\frac{1}{n+1}\sum_{j=1}^{n+1}{n+1% \choose j}\*\left(\prod_{p\mathbin{|}m}(1-p^{n-j})B_{n+1-j}\right)m^{j}.

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $j$ : integer, $k$ : integer, $m$ : integer, $n$ : integer and $p$ : prime
Referenced by:: §24.4(iii)
Permalink:: http://dlmf.nist.gov/24.4.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §24.4(iii), §24.4 and Ch.24

►

§24.4(iv) Finite Expansions

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24.4.24

B_{n}\left(mx\right)=m^{n}B_{n}\left(x\right)+n\sum_{k=1}^{n}\sum_{j=0}^{k-1}(% -1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}\frac{e^{2\pi i(k-j)r/m}}{(1-e^{2% \pi ir/m})^{n}}\right)(j+mx)^{n-1},

n=1,2,\dots

,

m=2,3,\dots

.

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Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $j$ : integer, $k$ : integer, $m$ : integer, $n$ : integer and $x$ : real or complex
Referenced by:: §24.4(v)
Permalink:: http://dlmf.nist.gov/24.4.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24

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§24.4(viii) Symbolic Operations

…

… ►where

x

is a spatial coordinate,

m

the mass of the particle with potential energy

V(x)

,

\hbar=h/(2\pi)

is the reduced Planck’s constant, and

(a,b)

a finite or infinite interval. … ►As in classical dynamics this sum is the total energy of the one particle system. … ►The finite system of functions

\psi_{n}

is orthonormal in

L^{2}\left(\mathbb{R},\,\mathrm{d}x\right)

, see (18.34.7_3). … ►

§18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods

… ►This equivalent quadrature relationship, see Heller et al. (1973), Yamani and Reinhardt (1975), allows extraction of scattering information from the finite dimensional

L^{2}

functions of (18.39.53), provided that such information involves potentials, or projections onto

L^{2}

functions, exactly expressed, or well approximated, in the finite basis of (18.39.44). …

finite sum of 3j symbols

11—20 of 23 matching pages

11: Errata

12: Bibliography F

13: 1.9 Calculus of a Complex Variable

14: Bibliography W

15: 3.7 Ordinary Differential Equations

16: 18.38 Mathematical Applications

$3j$ and $6j$ Symbols

17: 25.11 Hurwitz Zeta Function

§25.11(xi) Sums

18: 18.25 Wilson Class: Definitions

19: 24.4 Basic Properties

§24.4(iii) Sums of Powers

§24.4(iv) Finite Expansions

§24.4(viii) Symbolic Operations

20: 18.39 Applications in the Physical Sciences

§18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods

finite sum of 3j symbols

11—20 of 23 matching pages

11: Errata

12: Bibliography F

13: 1.9 Calculus of a Complex Variable

14: Bibliography W

15: 3.7 Ordinary Differential Equations

16: 18.38 Mathematical Applications

3 ⁢ j and 6 ⁢ j Symbols

17: 25.11 Hurwitz Zeta Function

§25.11(xi) Sums

18: 18.25 Wilson Class: Definitions

19: 24.4 Basic Properties

§24.4(iii) Sums of Powers

§24.4(iv) Finite Expansions

§24.4(viii) Symbolic Operations

20: 18.39 Applications in the Physical Sciences

§18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods

$3j$ and $6j$ Symbols