relation to minimax polynomials

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31: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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Chebyshev, Ultraspherical, and Jacobi

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§18.7(iii) Limit Relations

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Jacobi $\to$ Laguerre

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Laguerre $\to$ Hermite

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32: 7.18 Repeated Integrals of the Complementary Error Function

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§7.18(iv) Relations to Other Functions

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Hermite Polynomials

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Confluent Hypergeometric Functions

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Parabolic Cylinder Functions

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Probability Functions

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33: 25.13 Periodic Zeta Function

§25.13 Periodic Zeta Function

►The notation $F(x,s)$ is used for the polylogarithm ${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}x}\right)$ with $x$ real: ►

25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

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Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

… ►Also, … ►

25.13.3 $$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, (10), p. 257)

Notes:

This formula is equivalent to (25.11.9).

Referenced by:

Erratum (V1.1.4) for Equation (25.13.3)

Permalink:

http://dlmf.nist.gov/25.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

34: 10 Bessel Functions

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35: 7.11 Relations to Other Functions

§7.11 Relations to Other Functions

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Incomplete Gamma Functions and Generalized Exponential Integral

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7.11.1 $$\mathrm{erf}z=\frac{1}{\sqrt{\pi }}\gamma (\frac{1}{2},z^2),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\gamma (a,z)$: incomplete gamma function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions

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36: 5.23 Approximations

… ►Cody and Hillstrom (1967) gives minimax rational approximations for $\ln \mathrm{\Gamma }\left(x\right)$ for the ranges $0.5\le x\le 1.5$, $1.5\le x\le 4$, $4\le x\le 12$; precision is variable. Hart et al. (1968) gives minimax polynomial and rational approximations to $\mathrm{\Gamma }\left(x\right)$ and $\ln \mathrm{\Gamma }\left(x\right)$ in the intervals $0\le x\le 1$, $8\le x\le 1000$, $12\le x\le 1000$; precision is variable. Cody et al. (1973) gives minimax rational approximations for $\psi \left(x\right)$ for the ranges $0.5\le x\le 3$ and ; precision is variable. … ►See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of $\mathrm{\Gamma }\left(z\right)$. ►For rational approximations to $\psi \left(z\right)+\gamma$ see Luke (1975, pp. 13–16).

37: Bibliography I

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On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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External Links:

ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

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Special Functions, $q$-Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Mourad E.H. Ismail.

Encodings:

BibTeX

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:

ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

BibTeX

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Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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External Links:

ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(v).

Encodings:

BibTeX

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The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207

External Links:

ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§32.11(iii).

Encodings:

BibTeX

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38: 18.27 $q$-Hahn Class

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From Big $q$-Jacobi to Jacobi

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From Big $q$-Jacobi to Little $q$-Jacobi

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From Little $q$-Jacobi to Jacobi

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From Little $q$-Laguerre to Laguerre

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Limit Relations

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39: 18.35 Pollaczek Polynomials

… ►The type 2 polynomials reduce for $a=b=0$ to ultraspherical polynomials, see (18.35.8). ►The Pollaczek polynomials of type 3 are defined by the recurrence relation (in first form (18.2.8)) …the recurrence relation of form (18.2.11_5) becomes … ►As in the coefficients of the above recurrence relations $n$ and $c$ only occur in the form $n+c$, the type 3 Pollaczek polynomials may also be called the associated type 2 Pollaczek polynomials by using the terminology of §18.30. … ►we have the explicit representations …

40: William P. Reinhardt

… ►Reinhardt is a frequent visitor to the NIST Physics Laboratory in Gaithersburg, and to the Joint Quantum Institute (JQI) and Institute for Physical Sciences and Technology (ISTP) at the University of Maryland. … ►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. He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. … ►Reinhardt was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. …