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21: 31.15 Stieltjes Polynomials

… ►The $V(z)$ are called Van Vleck polynomials and the corresponding $S(z)$ Stieltjes polynomials. … ►

31.15.2 $$\sum _{j=1}^N\frac{{\gamma }_j/2}{z_k-a_j}+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^n\frac{1}{z_k-z_j}=0,$$ $k=1,2,\mathrm{\dots },n$.

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(ii)

Permalink:

http://dlmf.nist.gov/31.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

►If $t_k$ is a zero of the Van Vleck polynomial $V(z)$, corresponding to an $n$th degree Stieltjes polynomial $S(z)$, and $z_1^{\prime },z_2^{\prime },\mathrm{\dots },z_{n-1}^{\prime }$ are the zeros of $S^{\prime }(z)$ (the derivative of $S(z)$), then $t_k$ is either a zero of $S^{\prime }(z)$ or a solution of the equation … ►See Marden (1966), Alam (1979), and Al-Rashed and Zaheer (1985) for further results on the location of the zeros of Stieltjes and Van Vleck polynomials. …

22: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

BibTeX

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Guide to tables of elliptic functions. Math. Tables and Other Aids to Computation 3 (24), pp. 229–281.

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External Links:

FullText, MathReview (S. C. van Veen)

Referenced by:

§19.37(i).

Encodings:

BibTeX

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

BibTeX

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23: 30.10 Series and Integrals

… ►For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …

24: 36 Integrals with Coalescing Saddles

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25: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

26: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

27: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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28: 5.21 Methods of Computation

… ►For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). … ►For a comprehensive survey see van der Laan and Temme (1984, Chapter III). …

29: 27.2 Functions

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§27.2(i) Definitions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. … ►If $\left(a,n\right)=1$, then the Euler–Fermat theorem states that …

30: 19.35 Other Applications

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§19.35(i) Mathematical

►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …