relations%20to%20Lam%C3%A9%20polynomials

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11: 27.2 Functions

… ►(

\nu\left(1\right)

is defined to be 0.) Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …They tend to thin out among the large integers, but this thinning out is not completely regular. … ►the sum of the

k

th powers of the positive integers

m\leq n

that are relatively prime to

n

. … ►is the number of

k

-tuples of integers

\leq n

whose greatest common divisor is relatively prime to

n

. …

12: 26.5 Lattice Paths: Catalan Numbers

… ►

§26.5(i) Definitions

… ►It counts the number of lattice paths from

(0,0)

(n,n)

that stay on or above the line

y=x

. … ►

§26.5(iii) Recurrence Relations

… ►

26.5.6

C\left(n\right)\sim\frac{4^{n}}{\sqrt{\pi n^{3}}},

n\to\infty

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : 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

►

26.5.7

\lim_{n\to\infty}\frac{C\left(n+1\right)}{C\left(n\right)}=4.

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : 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: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

… ►For any partition

\kappa

, the zonal polynomial

Z_{\kappa}:\boldsymbol{\mathcal{S}}\to\mathbb{R}

is defined by the properties … ►

Normalization

… ►

Orthogonal Invariance

… ►

Summation

…

14: 6.11 Relations to Other Functions

§6.11 Relations to Other Functions

… ►

Incomplete Gamma Function

… ►

Confluent Hypergeometric Function

►

6.11.2

E_{1}\left(z\right)=e^{-z}U\left(1,1,z\right),

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : 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

►

6.11.3

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

ⓘ

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

15: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►

31.5.2

\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)=\mathit{H\!% \ell}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)

ⓘ

Defines:: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials
Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.5 and Ch.31

►is a polynomial of degree

n

, and hence a solution of (31.2.1) that is analytic at all three finite singularities

0,1,a

. These solutions are the Heun polynomials. …

16: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►

§26.4(i) Definitions

… ►It is also the number of

k

-dimensional lattice paths from

(0,0,\ldots,0)

(n_{1},n_{2},\ldots,n_{k})

. For

k=0,1

, the multinomial coefficient is defined to be

1

. … ►(The empty set is considered to have one permutation consisting of no cycles.) … ►

§26.4(iii) Recurrence Relation

…

17: 6.16 Mathematical Applications

… ►Hence, if

x

is fixed and

n\to\infty

, then

S_{n}(x)\to\frac{1}{4}\pi

0

, or

-\frac{1}{4}\pi

according as

0<x<\pi

x=0

, or

-\pi<x<0

; compare (6.2.14). … ►Hence if

x=\pi/(2n)

and

n\to\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

\tfrac{1}{2}

(§25.10(i)), then …where

\pi(x)

is the number of primes less than or equal to

x

. … ►

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

18: 8.26 Tables

… ►

•

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

… ►

•

Pearson (1965) tabulates the function $I(u,p)$ ( $=P\left(p+1,u\right)$ ) 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.

… ►

•

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 $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

•

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

… ►

•

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.

19: Bibliography I

… ►

K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

… ►

M. E. H. Ismail, D. R. Masson, and M. Rahman (Eds.) (1997) Special Functions,

q

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

ⓘ

External Links:: ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: Profile Mourad E.H. Ismail.
Encodings:: BibTeX

►

M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

ⓘ

External Links:: ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt
Referenced by:: §18.30(i).
Encodings:: BibTeX

►

M. E. H. Ismail and M. E. Muldoon (1995) Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

ⓘ

External Links:: ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt
Referenced by:: §10.21(v).
Encodings:: BibTeX

… ►

A. R. Its and A. A. Kapaev (1987) 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).

ⓘ

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

…

20: 14 Legendre and Related Functions

Chapter 14 Legendre and Related Functions

…