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1: Bibliography J

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Fonctions de Mathieu et polynômes de Klein-Gordon. C. R. Acad. Sci. Paris Sér. I Math. 325 (7), pp. 713–716 (French).

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Notes:

Translated title: Mathieu functions and Klein-Gordon polynomials.

External Links:

ISSN 0764-4442, Document, MathReview, ZentralBlatt

Referenced by:

7th item.

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Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D 1 (1), pp. 80–158.

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External Links:

ISSN 0167-2789, Document, MathReview, ZentralBlatt

Referenced by:

§32.16.

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BibTeX

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Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.

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External Links:

ISSN 0176-4276, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§18.24, §2.4(vi).

Encodings:

BibTeX

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Note sur la série ${\sum }_{n=1}^{\mathrm{\infty }}{x}^n/n^s$ . Bull. Soc. Math. France 17, pp. 142–152 (French).

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External Links:

ISSN 0037-9484, MathReview Entry, ZentralBlatt

Referenced by:

§25.12(ii).

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Memoire sur l’itération des fonctions rationnelles. J. Math. Pures Appl. 8 (1), pp. 47–245 (French).

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External Links:

FullText, ZentralBlatt

Referenced by:

§3.8(viii).

Encodings:

BibTeX

2: Bibliography D

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On the zeros of generalized Bessel polynomials. I. Nederl. Akad. Wetensch. Indag. Math. 84 (1), pp. 1–13.

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External Links:

ISSN 0019-3577, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.49(ii), §10.58.

Encodings:

BibTeX

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Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2. American Mathematical Society, Providence, RI.

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Notes:

Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997

External Links:

ISBN 0-8218-1198-3, MathReview, ZentralBlatt

Referenced by:

§21.9.

Encodings:

BibTeX

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The Bose-Einstein integrals ${ℬ}_p(\eta )={(p!)}^{-1}{\int }_0^{\mathrm{\infty }}{ϵ}^p{(e^{ϵ-\eta }-1)}^{-1}𝑑ϵ$ . Appl. Sci. Res. B. 6, pp. 240–244.

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External Links:

ISSN 0003-6994, MathReview Entry, ZentralBlatt

Referenced by:

(25.12.15), §25.12(iii), §25.12.

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The Fermi-Dirac integrals ${ℱ}_p(\eta )={(p!)}^{-1}{\int }_0^{\mathrm{\infty }}{ϵ}^p{(e^{ϵ-\eta }+1)}^{-1}𝑑ϵ$ . Appl. Sci. Res. B. 6, pp. 225–239.

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External Links:

ISSN 0003-6994, MathReview Entry, ZentralBlatt

Referenced by:

(25.12.14), §25.12(iii), §25.12.

Encodings:

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Infinite integrals in the theory of Bessel functions. Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.32(iii).

Encodings:

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3: 26.8 Set Partitions: Stirling Numbers

… ► $s(n,k)$ denotes the Stirling number of the first kind: ${(-1)}^{n-k}$ times the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with exactly $k$ cycles. … ►where ${\left(x\right)}_n$ is the Pochhammer symbol: $x(x+1)\mathrm{\cdots }(x+n-1)$. … ►For $n\ge 1$, … ►uniformly for $n=o\left(k^{1/2}\right)$. ►For asymptotic approximations for $s(n+1,k+1)$ and $S(n,k)$ that apply uniformly for $1\le k\le n$ as $n\to \mathrm{\infty }$ see Temme (1993) and Temme (2015, Chapter 34). …

4: 26.13 Permutations: Cycle Notation

… ► ${𝔖}_n$ denotes the set of permutations of $\{1,2,\mathrm{\dots },n\}$. … ►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. …In consequence, (26.13.2) can also be written as $\left(1,3,2,5,7\right)\left(6,8\right)$. … ►A permutation that consists of a single cycle of length $k$ can be written as the composition of $k-1$ two-cycles (read from right to left): … ►If , then $\left(j,k\right)$ is a product of $2k-2j-1$ adjacent transpositions: …

5: 26.1 Special Notation

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$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
$\left(\genfrac{}{}{0.0pt}{}{m}{n_1,n_2,\mathrm{\dots },n_k}\right)$	multinomial coefficient.
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$s(n,k)$	Stirling numbers of the first kind.
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$$x^{\overline{n}}=x(x+1)(x+2)\mathrm{\cdots }(x+n-1),$$

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$$x^{\underset{¯}{n}}=x(x-1)(x-2)\mathrm{\cdots }(x-n+1).$$

►Other notations for $s(n,k)$, the Stirling numbers of the first kind, include $S_n^{(k)}$ (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), $S_n^k$ (Jordan (1939), Moser and Wyman (1958a)), $\left(\genfrac{}{}{0.0pt}{}{n-1}{k-1}\right)B_{n-k}^{(n)}$ (Milne-Thomson (1933)), ${(-1)}^{n-k}{S}_1(n-1,n-k)$ (Carlitz (1960), Gould (1960)), ${(-1)}^{n-k}\left[\genfrac{}{}{0pt}{}{n}{k}\right]$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). …

6: 3.4 Differentiation

… ►The Lagrange $(n+1)$-point formula is … ►where ${\xi }_0$ and ${\xi }_1\in I$. ►For the values of $n_0$ and $n_1$ used in the formulas below … ►With the choice $r=k$ (which is crucial when $k$ is large because of numerical cancellation) the integrand equals ${\mathrm{e}}^k$ at the dominant points $\theta =0,2\pi$, and in combination with the factor $k^{-k}$ in front of the integral sign this gives a rough approximation to $1/k!$. … ►

3.4.34 $${\nabla }^4{u}_{0,0}=\frac{1}{6h^4}(184u_{0,0}-(u_{0,3}+u_{0,-3}+u_{3,0}+u_{-3,0})+14(u_{0,2}+u_{0,-2}+u_{2,0}+u_{-2,0})-77(u_{0,1}+u_{0,-1}+u_{1,0}+u_{-1,0})+20(u_{1,1}+u_{1,-1}+u_{-1,1}+u_{-1,-1})-(u_{1,2}+u_{2,1}+u_{1,-2}+u_{2,-1}+u_{-1,2}+u_{-2,1}+u_{-1,-2}+u_{-2,-1}))+O\left(h^4\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding and $u$: function

A&S Ref:

25.3.33

Permalink:

http://dlmf.nist.gov/3.4.E34

Encodings:

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See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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7: DLMF Project News

error generating summary

8: 24.15 Related Sequences of Numbers

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24.15.4 $$T_{2n-1}={(-1)}^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}{B}_{2n},$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

$B_n$: Bernoulli numbers, $n$: integer and $T_n$: tangent numbers

Referenced by:

§24.19(i)

Permalink:

http://dlmf.nist.gov/24.15.E4

Encodings:

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See also:

Annotations for §24.15(ii), §24.15 and Ch.24

… ►The Stirling numbers of the first kind $s(n,m)$, and the second kind $S(n,m)$, are as defined in §26.8(i). … ►

24.15.8 $$\sum _{k=0}^n{(-1)}^{n+k}s(n+1,k+1)B_k=\frac{n!}{n+1}.$$

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Symbols:

$B_n$: Bernoulli numbers, $s(n,k)$: Stirling number of the first kind, $!$: factorial (as in $n!$), $k$: integer and $n$: integer

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E8

Encodings:

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See also:

Annotations for §24.15(iii), §24.15 and Ch.24

… ►The Fibonacci numbers are defined by $u_0=0$, $u_1=1$, and $u_{n+1}=u_n+u_{n-1}$, $n\ge 1$. The Lucas numbers are defined by $v_0=2$, $v_1=1$, and $v_{n+1}=v_n+v_{n-1}$, $n\ge 1$. …

9: 17.18 Methods of Computation

… ►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. Method (1) is applicable within the circles of convergence of the defining series, although it is often cumbersome owing to slowness of convergence and/or severe cancellation. … ►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. …

10: 4.45 Methods of Computation

… ►Suppose first $1/10\le x\le 10$. …After computing $\ln \left(1+y\right)$ from (4.6.1) … ►For other values of $x$ set $x={10}^m\xi$, where $1/10\le \xi \le 10$ and $m\in \mathbb{Z}$. … ►From (4.24.15) with $u=v=({(1+x^2)}^{1/2}-1)/x$, we have … ►For example, $\arcsin x=\arctan \left(x{(1-x^2)}^{-1/2}\right)$. …