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21: 26.5 Lattice Paths: Catalan Numbers

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§26.5(i) Definitions

… ►It counts the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$. … ►

§26.5(iii) Recurrence Relations

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26.5.6 $$C\left(n\right)\sim \frac{4^n}{\sqrt{\pi n^3}},$$ $n\to \mathrm{\infty }$,

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Symbols:

$C\left(n\right)$: Catalan number, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E6

Encodings:

pMML, png, TeX

See also:

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

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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

22: 6.11 Relations to Other Functions

§6.11 Relations to Other Functions

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Incomplete Gamma Function

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Confluent Hypergeometric Function

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6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

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6.11.3 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)=U(1,1,-\mathrm{i}z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, 5th item

Permalink:

http://dlmf.nist.gov/6.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

23: Bibliography I

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

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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24: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

Encodings:

BibTeX

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Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§13.28(i).

Encodings:

BibTeX

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

BibTeX

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

BibTeX

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25: 6.16 Mathematical Applications

… ►Hence, if $x$ is fixed and $n\to \mathrm{\infty }$, then $S_n(x)\to \frac{1}{4}\pi$, $0$, or $-\frac{1}{4}\pi$ according as , $x=0$, or ; compare (6.2.14). … ►Hence if $x=\pi /(2n)$ and $n\to \mathrm{\infty }$, then the limiting value of $S_n(x)$ overshoots $\frac{1}{4}\pi$ by approximately 18%. … ►If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta \left(s\right)$ have real part of $\frac{1}{2}$ (§25.10(i)), then …where $\pi (x)$ is the number of primes less than or equal to $x$. … ►

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Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

26: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Pearson (1965) tabulates the function $I(u,p)$ ($=P(p+1,u)$) for $p=-1(.05)0(.1)5(.2)50$, $u=0(.1)u_p$ to 7D, where $I(u,u_p)$ rounds off to 1 to 7D; also $I(u,p)$ for $p=-0.75(.01)-1$, $u=0(.1)6$ to 5D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

27: 36.5 Stokes Sets

… ►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►The first sheet corresponds to and is generated as a solution of Equations (36.5.6)–(36.5.9). … ►This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the $z$-axis by $2\pi /3$. … ►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …

28: 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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Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:

Fortran, minimum precision 6D.

External Links:

ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

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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29: 26.9 Integer Partitions: Restricted Number and Part Size

… ►Conjugation establishes a one-to-one correspondence between partitions of $n$ into at most $k$ parts and partitions of $n$ into parts with largest part less than or equal to $k$. It follows that $p_k\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$. … ►

§26.9(iii) Recurrence Relations

… ►where the inner sum is taken over all positive divisors of $t$ that are less than or equal to $k$. … ►As $n\to \mathrm{\infty }$ with $k$ fixed, …

30: Publications

DLMF Related Publications

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Using Numerical Grid Generation to Facilitate 3D Visualization of Complicated Mathematical Functions, Technical Report NISTIR 6413 (November 1999), National Institute of Standards and Technology.

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From 2D to 3D: Numerical Grid Generation and the Visualization of Complex Surfaces, Proceedings of the 7th International Conference on Numerical Grid Generation in Computational Field Simulations, Whistler, British Columbia, Canada, September 25-28, 2000.

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From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243.

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NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53.