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1: 37.2 General Orthogonal Polynomials of Two Variables

… ►Then … ►Then necessarily ${⟨P_{k,n},Q_{k,n}⟩}_W\ne 0$ ($k=0,1,\mathrm{\dots },n$). … ► … ►

§37.2(v) Product Weight Functions

… ►

§37.2(vii) Rotation Invariant Weight Functions

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2: 18.32 OP’s with Respect to Freud Weights

§18.32 OP’s with Respect to Freud Weights

►A Freud weight is a weight function of the form ►

18.32.1 $$w(x)=\exp \left(-Q(x)\right),$$ ,

ⓘ

Symbols:

$\exp z$: exponential function, $w(x)$: weight function and $x$: real variable

Referenced by:

§18.32

Permalink:

http://dlmf.nist.gov/18.32.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.32 and Ch.18

… ►For asymptotic approximations to OP’s that correspond to Freud weights with more general functions $Q(x)$ see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999). … ►

18.32.2 $$w(x)={\left|x\right|}^{\alpha }\exp \left(-Q(x)\right),$$ $x\in ℝ$, $\alpha >-1$,

ⓘ

Symbols:

$\in$: element of, $\exp z$: exponential function, $ℝ$: real line, $w(x)$: weight function and $x$: real variable

Referenced by:

§18.32, Erratum (V1.2.0) §18.32

Permalink:

http://dlmf.nist.gov/18.32.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.32 and Ch.18

…

3: 37.10 Other Orthogonal Polynomials of Two Variables

… ►For the weight function … ► … ►The Bernstein–Szegő weight function is defined by …One example of the weight function is …The OPs for these weight functions can be constructed explicitly and they are studied in Delgado et al. (2009). …

4: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►These are orthogonal polynomials for a family of reflection invariant weight functions on the unit sphere. …The weight function is invariant under the reflection group $G$. … ►

§37.19(ii) OPs on the Ball for Generalized Weight Functions

… ►

§37.19(iii) OPs on ${ℝ}^d$ for Generalized Weight Functions

►Let $w_{\kappa }$ be the weight function (37.19.5). …

5: 37.13 General Orthogonal Polynomials of $d$ Variables

… ►Define an inner product … ►

§37.13(i) OPs for a Rotation Invariant Weight Function

… ►For each $k\in {\mathbb{N}}_0$ let $p_n^{(k)}(x)$ denote an OP of degree $n$ for the weight function $x^{k}w(x)$ on $[0,\mathrm{\infty })$. The space ${𝒱}_n^d$ for the rotation invariant weight function $W$ can be orthogonally decomposed as … ►and weight function …

6: 12.15 Generalized Parabolic Cylinder Functions

… ►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …

7: null

error generating summary

8: 37.5 Quarter Plane with Weight Function $x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y}$

§37.5 Quarter Plane with Weight Function $x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y}$

… ►define the weight function … ►Define Laguerre–Jacobi polynomials on the quarter plane by … ►

§37.5(ii) Differential Equations

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§37.5(iii) Limits

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9: 18.31 Bernstein–Szegő Polynomials

… ►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: ${(1-x^2)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x^2)}^{\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x)}^{\frac{1}{2}}{(1+x)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$. …

10: 37.7 Parabolic Biangular Region with Weight Function ${\left(1-x\right)}^{\alpha }{\left(x-y^2\right)}^{\beta }$

§37.7 Parabolic Biangular Region with Weight Function ${\left(1-x\right)}^{\alpha }{\left(x-y^2\right)}^{\beta }$

… ►bounded by a parabolic arc and a line segment, define the weight function … ►

§37.7(ii) Quadratic Transformations

… ►

§37.7(iii) Differential Equations

… ►The polynomials (37.3.9) and (37.7.16) are related by the quadratic transformations …