q-Laguerre polynomials

SearchAdvancedHelp

(0.002 seconds)

41—50 of 255 matching pages

41: 29.14 Orthogonality

§29.14 Orthogonality

►Lamé polynomials are orthogonal in two ways. … ►

29.14.4 $${\mathrm{𝑠𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑠𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

►

29.14.5 $${\mathrm{𝑐𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑐𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

►

29.14.6 $${\mathrm{𝑑𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑑𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

…

42: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►For an asymptotic expansion of $P_n^{(\lambda )}(nx;\varphi )$ as $n\to \mathrm{\infty }$, with $\varphi$ fixed, see Li and Wong (2001). …Corresponding approximations are included for the zeros of $P_n^{(\lambda )}(nx;\varphi )$. ►

Approximations in Terms of Laguerre Polynomials

… ►Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.

43: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

►For orthogonal polynomials see Chapter 18. …

44: 32.15 Orthogonal Polynomials

§32.15 Orthogonal Polynomials

►Let $p_n(\xi )$, $n=0,1,\mathrm{\dots }$, be the orthonormal set of polynomials defined by ►

32.15.1 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(-\frac{1}{4}{\xi }^4-z{\xi }^2\right)p_m(\xi )p_n(\xi )d\xi ={\delta }_{m,n},$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $m$: integer, $n$: integer, $z$: real and $p_n(\xi )$: polynomials

Permalink:

http://dlmf.nist.gov/32.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.15 and Ch.32

… ►

32.15.2 $$a_{n+1}(z)p_{n+1}(\xi )=\xi p_n(\xi )-a_n(z)p_{n-1}(\xi ),$$

ⓘ

Symbols:

$n$: integer, $z$: real and $p_n(\xi )$: polynomials

Permalink:

http://dlmf.nist.gov/32.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §32.15 and Ch.32

… ►

45: 35.12 Software

… ►

Demmel and Koev (2006). Computation of zonal polynomials in MATLAB.

… ►

Stembridge (1995). Maple software for zonal polynomials.

►For an algorithm to evaluate zonal polynomials, and an implementation of the algorithm in Maple by Zeilberger, see Lapointe and Vinet (1996).

46: Roelof Koekoek

… ►Koekoek is mainly a teacher of mathematics and has published a few papers on orthogonal polynomials. He is also author of the book Hypergeometric Orthogonal Polynomials and Their $q$-Analogues (with P. … ►

18 Orthogonal Polynomials

…

47: 18.27 $q$-Hahn Class

… ►

§18.27(ii) $q$-Hahn Polynomials

… ►

§18.27(iii) Big $q$-Jacobi Polynomials

… ►

§18.27(iv) Little $q$-Jacobi Polynomials

… ►

Little $q$-Laguerre polynomials

… ►

§18.27(v) $q$-Laguerre Polynomials

…

48: 18.31 Bernstein–Szegő Polynomials

§18.31 Bernstein–Szegő Polynomials

►Let $\rho (x)$ be a polynomial of degree $\mathrm{\ell }$ and positive when $-1\le x\le 1$. The Bernstein–Szegő polynomials $\{p_n(x)\}$, $n=0,1,\mathrm{\dots }$, are orthogonal on $(-1,1)$ with respect to three types of weight function: ${(1-x^2)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x^2)}^{\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x)}^{\frac{1}{2}}{(1+x)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$. …

49: 24.2 Definitions and Generating Functions

… ►

§24.2(i) Bernoulli Numbers and Polynomials

… ►

§24.2(ii) Euler Numbers and Polynomials

… ►

${(-1)}^n{E}_{2n}>0$ .

… ►

Table 24.2.2: Bernoulli and Euler polynomials.

► ►►

$n$	$B_n\left(x\right)$	$E_n\left(x\right)$
…

►

…

50: 18.22 Hahn Class: Recurrence Relations and Differences

… ►

§18.22(i) Recurrence Relations in $n$

… ►These polynomials satisfy (18.22.2) with $p_n(x)$, $A_n$, and $C_n$ as in Table 18.22.1. ►

Table 18.22.1: Recurrence relations (18.22.2) for Krawtchouk, Meixner, and Charlier polynomials.

► ►►

$p_n(x)$	$A_n$	$C_n$
…

►

… ►

§18.22(ii) Difference Equations in $x$

… ►

§18.22(iii) $x$-Differences

…