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11: Errata

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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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Subsection 5.2(iii)

Three new identities for Pochhammer’s symbol (5.2.6)–(5.2.8) have been added at the end of this subsection.

Suggested by Tom Koornwinder.

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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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12: 18.35 Pollaczek Polynomials

… ►The three types of Pollaczek polynomials were successively introduced in Pollaczek (1949a, b, 1950), see also Erdélyi et al. (1953b, p.219) and, for type 1 and 2, Szegö (1950) and Askey (1982b). … ►

18.35.2_2

{Q}^{{(\lambda)}}_{n}\left(x;a,b,c\right)=\frac{{\left(c+1\right)_{n}}}{2^{n}{% \left(c+\lambda+a\right)_{n}}}\,P^{{(\lambda)}}_{n}\left(x;a,b,c\right)

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Defines:: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : Pollaczek polynomial
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.35(ii)
Permalink:: http://dlmf.nist.gov/18.35.E2_2
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.35(i), §18.35 and Ch.18

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18.35.4_5

P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)=\frac{{\left(2\lambda\right)_{n}}% }{n!}\,{\mathrm{e}}^{\mathrm{i}n\theta}\*F\left({-n,\lambda+\mathrm{i}\tau_{a,% b}(\theta)\atop 2\lambda};1-{\mathrm{e}}^{-2\mathrm{i}\theta}\right).

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$ : Pollaczek polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $n$ : nonnegative integer and $\tau_{a,b}(\theta)$
Source:: Erdélyi et al. (1953b, 10.21(10))
Proof sketch:: Apply (15.8.7) to (18.35.4).
Referenced by:: (18.35.9), §18.35(i), Erratum (V1.2.0) Section 18.35
Permalink:: http://dlmf.nist.gov/18.35.E4_5
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.35(i), §18.35 and Ch.18

… ►More generally, the

P^{(\lambda)}_{n}\left(x;a,b\right)

are OP’s if and only if one of the following three conditions holds (in case (iii) work with the monic polynomials (18.35.2_2)). … ►

18.35.6_5

{\int_{-1}^{1}P^{{(\lambda)}}_{n}\left(x;a,b,c\right)P^{{(\lambda)}}_{m}\left(% x;a,b,c\right)w^{(\lambda)}(x;a,b,c)\,\mathrm{d}x=\frac{\Gamma\left(c+1\right)% \Gamma\left(2\lambda+c+n\right)}{{\left(c+1\right)_{n}}(\lambda+a+c+n)}\,% \delta_{n,m},}

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : Pollaczek polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $w(x)$ : weight function, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Proved:: Pollaczek (1950, (3))(proved)
Permalink:: http://dlmf.nist.gov/18.35.E6_5
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.35(ii), §18.35 and Ch.18

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13: 18.27 $q$ -Hahn Class

… ►The generic (top level) cases are the

q

-Hahn polynomials and the big

q

-Jacobi polynomials, each of which depends on three further parameters. … ►

18.27.4

\sum_{y=0}^{N}Q_{n}(q^{-y})Q_{m}(q^{-y})\genfrac{[}{]}{0.0pt}{}{N}{y}_{q}\frac% {\left(\alpha q;q\right)_{y}\left(\beta q;q\right)_{N-y}}{\left(\alpha q\right% )^{y}}=h_{n}\delta_{n,m},

n,m=0,1,\ldots,N

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $y$ : real variable, $N$ : positive integer, $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $h_{n}$
Notes:: This is Ismail (2009, (18.5.2)) but the presentation has been simplified.
Referenced by:: §18.27(ii), Erratum (V1.2.0) for Equation (18.27.4)
Permalink:: http://dlmf.nist.gov/18.27.E4
Encodings:: pMML, png, TeX
Correction (effective with 1.1.2):: $q$ -Pochhammer symbols now link to their definition.
See also:: Annotations for §18.27(ii), §18.27 and Ch.18

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18.27.14

\sum_{y=0}^{\infty}p_{n}(q^{y})p_{m}(q^{y})\frac{\left(bq;q\right)_{y}(aq)^{y}% }{\left(q;q\right)_{y}}=h_{n}\delta_{n,m},

0<a<q^{-1},b<q^{-1}

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $y$ : real variable, $p_{n}(x)$ : polynomial of degree $n$ , $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.27.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §18.27(iv), §18.27 and Ch.18

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18.27.19

\int_{0}^{\infty}\frac{S_{n}\left(x;q\right)S_{m}\left(x;q\right)}{\left(-x,-% qx^{-1};q\right)_{\infty}}\,\mathrm{d}x=\frac{\ln\left(q^{-1}\right)}{q^{n}}% \frac{\left(q;q\right)_{\infty}}{\left(q;q\right)_{n}}\delta_{n,m},

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$ : Stieltjes–Wigert polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.27.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §18.27(vi), §18.27 and Ch.18

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18.27.22

\sum_{\ell=0}^{\infty}\left(h_{n}\left(q^{\ell};q\right)h_{m}\left(q^{\ell};q% \right)+h_{n}\left(-q^{\ell};q\right)h_{m}\left(-q^{\ell};q\right)\right)\*% \left(q^{\ell+1},-q^{\ell+1};q\right)_{\infty}q^{\ell}=\left(q;q\right)_{n}% \left({q,-1,-q};q\right)_{\infty}q^{n(n-1)/2}\delta_{n,m}.

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $h_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$ : discrete $q$ -Hermite I polynomial, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable, $\ell$ : nonnegative integer, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.27.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §18.27(vii), §18.27(vii), §18.27 and Ch.18

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14: Bibliography T

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J. G. Taylor (1982) Improved error bounds for the Liouville-Green (or WKB) approximation. J. Math. Anal. Appl. 85 (1), pp. 79–89.

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External Links:: ISSN 0022-247X, Document, MathReview (Klaus Deimling), ZentralBlatt
Referenced by:: §2.7(iii).
Encodings:: BibTeX

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N. M. Temme and J. L. López (2001) The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis. J. Comput. Appl. Math. 133 (1-2), pp. 623–633.

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External Links:: ISSN 0377-0427, Document, MathReview (P. N. Rathie), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

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N.M. Temme and E.J.M. Veling (2022) Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z. Indagationes Mathematicae.

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External Links:: ISSN 0019-3577, Document, Link
Referenced by:: §13.8(iv).
Encodings:: BibTeX

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W. J. Thompson (1994) Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

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Notes:: With 1 Macintosh floppy disk (3.5 inch; DD). Programs for the calculation of $3j,6j,9j$ symbols are included.
External Links:: ISBN 0-471-55264-X, MathReview (J. F. Cornwell), ZentralBlatt
Referenced by:: §34.3(vii).
Encodings:: BibTeX

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J. Todd (1954) Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.

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External Links:: MathReview (H. Bückner), ZentralBlatt
Referenced by:: §6.18(i).
Encodings:: BibTeX

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15: 19.29 Reduction of General Elliptic Integrals

… ►There are only three distinct

U

’s with subscripts

\leq 4

, and at most one of them can be 0 because the

d

’s are nonzero. … ►The advantages of symmetric integrals for tables of integrals and symbolic integration are illustrated by (19.29.4) and its cubic case, which replace the

8+8+12=28

formulas in Gradshteyn and Ryzhik (2000, 3.147, 3.131, 3.152) after taking

x^{2}

as the variable of integration in 3. … ►where

\mathbf{e}_{j}

is an

n

-tuple with 1 in the

j

th position and 0’s elsewhere. … ►Next, for

j=1,2

, define

Q_{j}(t)=f_{j}+g_{j}t+h_{j}t^{2}

, and assume both

Q

’s are positive for

y<t<x

. …where …

16: Mathematical Introduction

… ►The first three chapters of the NIST Handbook and DLMF are methodology chapters that provide detailed coverage of, and references for, mathematical topics that are especially important in the theory, computation, and application of special functions. … ► ►►►

$\mathbb{C}$	complex plane (excluding infinity).
…
$\delta_{j,k}$ or $\delta_{jk}$	Kronecker delta: 0 if $j\neq k$ ; 1 if $j=k$ .
…

► ►►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$[a_{j,k}]$ or $[a_{jk}]$	matrix with $(j,k)$ th element $a_{j,k}$ or $a_{jk}$ .
…
${\left(\alpha\right)_{n}}$	Pochhammer’s symbol: $\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+n-1)$ if $n=1,2,3,\dotsc$ ; 1 if $n=0$ .
…

… ► J. …

17: Bibliography D

… ►

A. Deaño, J. Segura, and N. M. Temme (2010) Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.

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External Links:: ISSN 0377-0427, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §13.29(iv), §13.29(iv).
Encodings:: BibTeX

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Derive (commercial interactive system) Texas Instruments, Inc..

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Notes:: Symbolic and extended-precision numerical computing system.
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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A. Dienstfrey and J. Huang (2006) Integral representations for elliptic functions. J. Math. Anal. Appl. 316 (1), pp. 142–160.

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External Links:: ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt
Referenced by:: §23.11.
Encodings:: BibTeX

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J. J. Duistermaat (1974) Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.

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External Links:: MathReview (Alan Weinstein), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

… ►

B. I. Dunlap and B. R. Judd (1975) Novel identities for simple

n

j

symbols. J. Mathematical Phys. 16, pp. 318–319.

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External Links:: Document, MathReview (M. J. Westwater)
Referenced by:: §34.5(vi).
Encodings:: BibTeX

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18: 18.39 Applications in the Physical Sciences

… ►Here are three examples of solutions for (18.39.8) for explicit choices of

V(x)

and with the

\psi_{n}(x)

corresponding to the discrete spectrum. All are written in the same form as the product of three factors: the square root of a weight function

w(x)

, the corresponding OP or EOP, and constant factors ensuring unit normalization. … ►

§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom

… ►

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

… ►As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics. …