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11: 23 Weierstrass Elliptic and Modular
Functions

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12: 36 Integrals with Coalescing Saddles

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13: Peter L. Walker

… ► 1942 in Dorchester, U. …He began his academic career in 1964 at the University of Lancaster, U. … ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►Walker is now retired and living in Cheltenham, UK. … ►

20 Theta Functions

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14: William P. Reinhardt

… ►He currently resides in Santa Fe, NM. ►Reinhardt is a theoretical chemist and atomic physicist, who has always been interested in orthogonal polynomials and in the analyticity properties of the functions of mathematical physics. … ►This is closely connected with his interests in classical dynamical “chaos,” an area where he coauthored a book, Chaos in atomic physics with Reinhold Blümel. … ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

15: 27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product $m$ has 20 digits, twice the number of digits in the data. By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. … ►Details of a machine program describing the method together with typical numerical results can be found in Newman (1967). …

16: 18.20 Hahn Class: Explicit Representations

… ►For comments on the use of the forward-difference operator ${\mathrm{\Delta }}_x$, the backward-difference operator ${\nabla }_x$, and the central-difference operator ${\delta }_x$, see §18.2(ii). … ►In (18.20.1) $X$ and $w_x$ are as in Table 18.19.1. …For the Krawtchouk, Meixner, and Charlier polynomials, $F(x)$ and ${\kappa }_n$ are as in Table 18.20.1. … ►

18.20.3 $$w(x;a,b,\overline{a},\overline{b})p_n(x;a,b,\overline{a},\overline{b})=\frac{1}{n!}{\delta }_x^n\left(w(x;a+\frac{1}{2}n,b+\frac{1}{2}n,\overline{a}+\frac{1}{2}n,\overline{b}+\frac{1}{2}n)\right).$$

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Symbols:

${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i)

Permalink:

http://dlmf.nist.gov/18.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

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18.20.4 $$w^{(\lambda )}(x;\varphi )P_n^{(\lambda )}(x;\varphi )=\frac{1}{n!}{\delta }_x^n\left(w^{(\lambda +\frac{1}{2}n)}(x;\varphi )\right).$$

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\delta }_x$: central difference, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i)

Permalink:

http://dlmf.nist.gov/18.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

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17: Bibliography C

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Algorithm AS 310. Computing the non-central beta distribution function. Appl. Statist. 46 (1), pp. 146–156.

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Notes:

Single-precision Fortran.

External Links:

FullText, Code, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

Encodings:

BibTeX

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Some non-central distribution problems in multivariate analysis. Ann. Math. Statist. 34 (4), pp. 1270–1285.

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External Links:

FullText, MathReview (I. Olkin), ZentralBlatt

Referenced by:

§35.4(i), §35.4(ii).

Encodings:

BibTeX

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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:

Document

Referenced by:

5th item.

Encodings:

BibTeX

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18: 18.39 Applications in the Physical Sciences

§18.39 Applications in the Physical Sciences

… ►The spectrum is entirely discrete as in §1.18(v). … ►In what follows the radial and spherical radial eigenfunctions corresponding to (18.39.27) are found in four different notations, with identical eigenvalues, all of which appear in the current and past mathematical and theoretical physics and chemistry literatures, regarding this central problem. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … ►

§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences

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19: 18.22 Hahn Class: Recurrence Relations and Differences

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§18.22(i) Recurrence Relations in $n$

… ►These polynomials satisfy (18.22.2) with $p_n(x)$, $A_n$, and $C_n$ as in Table 18.22.1. … ►

§18.22(ii) Difference Equations in $x$

… ►For $A(x)$, $C(x)$, and ${\lambda }_n$ in (18.22.12) see Table 18.22.2. … ►

18.22.27 $${\delta }_x\left(p_n(x;a,b,\overline{a},\overline{b})\right)=(n+2\mathrm{\Re }\left(a+b\right)-1)p_{n-1}(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2}),$$

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Symbols:

${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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20: 10.75 Tables

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Achenbach (1986) tabulates $J_0\left(x\right)$, $J_1\left(x\right)$, $Y_0\left(x\right)$, $Y_1\left(x\right)$, $x=0(.1)8$, 20D or 18–20S.

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Bickley et al. (1952) tabulates $x^{-n}{I}_n\left(x\right)$ or ${\mathrm{e}}^{-x}{I}_n\left(x\right)$, $x^n{K}_n\left(x\right)$ or ${\mathrm{e}}^x{K}_n\left(x\right)$, $n=2(1)20$, $x=0$(.01 or .1) 10(.1) 20, 8S; $I_n\left(x\right)$, $K_n\left(x\right)$, $n=0(1)20$, $x=0$ or $0.1(.1)20$, 10S.

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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_n\left(z\right)$ and $K_n^{\prime }\left(z\right)$, for $n=2(1)20$, 9S.

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Kerimov and Skorokhodov (1984c) tabulates all zeros of $I_{-n-\frac{1}{2}}\left(z\right)$ and $I_{-n-\frac{1}{2}}^{\prime }\left(z\right)$ in the sector $0\le \mathrm{ph}z\le \frac{1}{2}\pi$ for $n=1(1)20$, 9S.

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The main tables in Abramowitz and Stegun (1964, Chapter 10) give ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$ $n=0(1)8$, $x=0(.1)10$, 5–8S; ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$ $n=0(1)20(10)50$, 100, $x=1,2,5,10,50,100$, 10S; ${𝗂}_n^{(1)}\left(x\right)$, ${𝗄}_n\left(x\right)$, $n=0,1,2$, $x=0(.1)5$, 4–9D; ${𝗂}_n^{(1)}\left(x\right)$, ${𝗄}_n\left(x\right)$, $n=0(1)20(10)50$, 100, $x=1,2,5,10,50,100$, 10S. (For the notation see §10.1 and §10.47(ii).)

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