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11: 3.4 Differentiation

… ►The $B_k^n$ are the differentiated Lagrangian interpolation coefficients: …where $A_k^n$ is as in (3.3.10). … ►

$$B_{-2}^6=\frac{1}{60}(9-9t-30t^2+20t^3+5t^4-3t^5),$$

… ►where $C$ is a simple closed contour described in the positive rotational sense such that $C$ and its interior lie in the domain of analyticity of $f$, and $x_0$ is interior to $C$. Taking $C$ to be a circle of radius $r$ centered at $x_0$, we obtain …

12: 26.5 Lattice Paths: Catalan Numbers

… ► $C\left(n\right)$ is the Catalan number. …(Sixty-six equivalent definitions of $C\left(n\right)$ are given in Stanley (1999, pp. 219–229).) … ►

26.5.3 $$C\left(n+1\right)=\sum _{k=0}^{n}C\left(k\right)C\left(n-k\right),$$

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Symbols:

$C\left(n\right)$: Catalan number, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

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26.5.4 $$C\left(n+1\right)=\frac{2(2n+1)}{n+2}C\left(n\right),$$

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Symbols:

$C\left(n\right)$: Catalan number and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

… ►

26.5.7 $$\underset{n\to \mathrm{\infty }}{lim}\frac{C\left(n+1\right)}{C\left(n\right)}=4.$$

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Symbols:

$C\left(n\right)$: Catalan number and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iv), §26.5 and Ch.26

13: 14.33 Tables

… ►

Abramowitz and Stegun (1964, Chapter 8) tabulates ${𝖯}_n\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 5–8D; ${𝖯}_n^{\prime }\left(x\right)$ for $n=1(1)4,9,10$, $x=0(.01)1$, 5–7D; ${𝖰}_n\left(x\right)$ and ${𝖰}_n^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 6–8D; $P_n\left(x\right)$ and $P_n^{\prime }\left(x\right)$ for $n=0(1)5,9,10$, $x=1(.2)10$, 6S; $Q_n\left(x\right)$ and $Q_n^{\prime }\left(x\right)$ for $n=0(1)3,9,10$, $x=1(.2)10$, 6S. (Here primes denote derivatives with respect to $x$.)

►

Zhang and Jin (1996, Chapter 4) tabulates ${𝖯}_n\left(x\right)$ for $n=2(1)5,10$, $x=0(.1)1$, 7D; ${𝖯}_n\left(\cos \theta \right)$ for $n=1(1)4,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${𝖰}_n\left(x\right)$ for $n=0(1)2,10$, $x=0(.1)0.9$, 8S; ${𝖰}_n\left(\cos \theta \right)$ for $n=0(1)3,10$, $\theta =0(5^{\circ }){90}^{\circ }$, 8D; ${𝖯}_n^m\left(x\right)$ for $m=1(1)4$, $n-m=0(1)2$, $n=10$, $x=0,0.5$, 8S; ${𝖰}_n^m\left(x\right)$ for $m=1(1)4$, $n=0(1)2,10$, 8S; ${𝖯}_{\nu }^m\left(\cos \theta \right)$ for $m=0(1)3$, $\nu =0(.25)5$, $\theta =0({15}^{\circ }){90}^{\circ }$, 5D; $P_n\left(x\right)$ for $n=2(1)5,10$, $x=1(1)10$, 7S; $Q_n\left(x\right)$ for $n=0(1)2,10$, $x=2(1)10$, 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 $\nu$-zeros of ${𝖯}_{\nu }^m\left(\cos \theta \right)$ and of its derivative for $m=0(1)4$, $\theta ={10}^{\circ },{30}^{\circ },{150}^{\circ }$.

►

Belousov (1962) tabulates ${𝖯}_n^m\left(\cos \theta \right)$ (normalized) for $m=0(1)36$, $n-m=0(1)56$, $\theta =0({2.5}^{\circ }){90}^{\circ }$, 6D.

…

14: Bibliography P

… ►

A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.

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External Links:

Document, ZentralBlatt

Referenced by:

§29.18(iv).

Encodings:

BibTeX

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

Encodings:

BibTeX

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On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.

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External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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Primzahlverteilung. Die Grundlehren der mathematischen Wissenschaften, Vol. 91, Springer-Verlag, Berlin-Göttingen-Heidelberg (German).

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Notes:

Reprinted in 1978.

External Links:

MathReview (A. E. Ingham), ZentralBlatt

Referenced by:

§27.11.

Encodings:

BibTeX

… ►

Stacking models of vesicles and compact clusters. J. Statist. Phys. 80 (3–4), pp. 755–779.

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External Links:

Document, ZentralBlatt

Referenced by:

§8.24(ii).

Encodings:

BibTeX

…

15: Bibliography K

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Branch Cuts for Complex Elementary Functions or Much Ado About Nothing’s Sign Bit. In The State of the Art in Numerical Analysis (Birmingham, 1986), A. Iserles and M. J. D. Powell (Eds.), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 9, pp. 165–211.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§4.23(iv), §4.37(iv).

Encodings:

BibTeX

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Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.

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External Links:

ISSN 0022-2488, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(ii).

Encodings:

BibTeX

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Poly-Bernoulli numbers. J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.

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External Links:

ISSN 1246-7405, PostScript, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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Overview of some new results concerning the theory and applications of the Rayleigh special function. Comput. Math. Math. Phys. 48 (9), pp. 1454–1507.

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External Links:

ISSN 0965-5425, Document, FullText

Referenced by:

§10.21(xiii), §10.21(xiii).

Encodings:

BibTeX

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On the zeros of the Fresnel integrals. Canad. J. Math. 9, pp. 118–131.

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External Links:

MathReview (W. C. Rheinboldt), ZentralBlatt

Referenced by:

§7.13(iii).

Encodings:

BibTeX

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16: Bibliography L

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Exact computation of the $9$-$j$ symbols. Comput. Phys. Comm. 70 (3), pp. 544–556.

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Notes:

Double-precision Fortran

External Links:

ISSN 0010-4655, Document, Code, MathReview

Referenced by:

7th item.

Encodings:

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Statistical Physics, Part 2: Theory of the Condensed State. Pergamon Press, Oxford.

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Notes:

Course of Theoretical Physics , Vol. 9. Translated from the Russian by J. B. Sykes and M. J. Kearsley.

External Links:

ISBN 0-08-023073-3; 0-08-023072-5, MathReview

Referenced by:

§25.17.

Encodings:

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Asymptotics of the first Appell function $F_1$ with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

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Monotonicity of the differences of zeros of Bessel functions as a function of order. Proc. Amer. Math. Soc. 15 (1), pp. 91–96.

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External Links:

ISSN 0002-9939, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(ii).

Encodings:

BibTeX

… ►

Deep water ship-waves. Phil. Mag. 9, pp. 733–757.

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External Links:

ZentralBlatt

Referenced by:

§36.13.

Encodings:

BibTeX

…

17: 34 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3}j,\mathit{6}j,\mathit{9}j$ Symbols

…

18: 1.12 Continued Fractions

… ► $C_n$ is called the $n$th approximant or convergent to $C$ . $A_n$ and $B_n$ are called the $n$th (canonical) numerator and denominator respectively. … ►Define … ►Conversely, $C$ is called an extension of $C^{\prime }$. … ►Then the convergents $C_n$ satisfy …

19: 1.11 Zeros of Polynomials

… ►Set $z=w-\frac{1}{3}a$ to reduce $f(z)=z^3+a{z}^2+bz+c$ to $g(w)=w^3+pw+q$, with $p=(3b-a^2)/3$, $q=(2a^3-9ab+27c)/27$. … ► $f(z)=z^3-6z^2+6z-2$, $g(w)=w^3-6w-6$, $A=3\sqrt[3]{4}$, $B=3\sqrt[3]{2}$. … ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. … ►Let … ►Then $f(z)$, with $a_n\ne 0$, is stable iff $a_0\ne 0$; $D_{2k}>0$, $k=1,\mathrm{\dots },\lfloor \frac{1}{2}n\rfloor$; $\mathrm{sign}{D}_{2k+1}=\mathrm{sign}{a}_0$, $k=0,1,\mathrm{\dots },\lfloor \frac{1}{2}n-\frac{1}{2}\rfloor$.

20: Bibliography N

… ►

Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

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On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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External Links:

ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

§11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).

Encodings:

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On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

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Der Eulersche Dilogarithmus und seine Verallgemeinerungen. Nova Acta Leopoldina 90, pp. 123–212.

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External Links:

ZentralBlatt

Referenced by:

§25.12(i).

Encodings:

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The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

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Notes:

In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.

External Links:

ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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