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1: Mourad E. H. Ismail

… ►His well-known book Classical and Quantum Orthogonal Polynomials in One Variable was published by Cambridge University Press in 2005 and reprinted with corrections in paperback in Ismail (2009). …

2: 21.8 Abelian Functions

… ►In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. …

3: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►These are orthogonal polynomials for an inner product that involves derivatives of functions; see Marcellán and Xu (2015) for the one-variable case. … ►Just as the classical OPs fit into the Askey scheme (see §18.19 and Figure 18.21.1) with Wilson and Racah polynomials on top, the Jacobi polynomials on the simplex fit into a scheme of OPs defined as products of one-variable OPs belonging to the Askey scheme by formulas somewhat resembling (37.14.7). However, when the one-variable OPs are taken from a higher level in the Askey scheme, the analogues of the denominators in the arguments in (37.14.7) will be parameters depending on

x_{\ell}

variables. … ►Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials of several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. In one variable they are essentially ultraspherical, Jacobi, continuous

q

-ultraspherical, or Askey–Wilson polynomials. …

4: 37.20 Mathematical Applications

… ►For the unit ball and the simplex, these quantities can be written as an one-variable integral involving the Jacobi polynomials. …

5: Bibliography I

… ►

M. E. H. Ismail (2005) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt
Referenced by:: §1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

►

M. E. H. Ismail (2009) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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Notes:: With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.
External Links:: ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt
Referenced by:: §18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.
Encodings:: BibTeX

…

6: 37.2 General Orthogonal Polynomials of Two Variables

… ►In the other direction, as an analogue of Favard’s theorem (see §18.2(viii) for the one-variable case), any polynomial system that satisfies the three-term relations (37.2.7), together with the conditions (37.2.10) and (37.2.8) of the coefficient matrices, must be orthonormal with respect to a positive definite linear functional. … ►If the equality holds in (37.2.13), then the common zeros are nodes of Gaussian cubature rules that are a complete analogue of Gaussian quadrature in one variable (see §3.5(v)). … ►where

p_{k}(w_{\ell},x)

is the OP of degree

k

for the weight function

w_{\ell}(x)

on an interval in

\mathbb{R}

. … ►Also, let

\{q_{n}(x)\}_{n=0,1,2,\ldots}

be a system of OPs on

(0,1)

with respect to the weight function

w_{2}(x)

, and put … ►

7: 37.10 Other Orthogonal Polynomials of Two Variables

… ►

§37.10(ii) Orthogonal Polynomials on an Annulus

… ►See §18.32 for Bernstein–Szegő polynomials of one variable. … ►

37.10.7

\lim_{N\to\infty}Q_{k,n}(Nx,Ny;\alpha,\beta,\gamma,N)=\frac{P_{k,n}^{\alpha,% \beta,\gamma}(x,y)}{P_{k,n}^{\alpha,\beta,\gamma}(0,0)}.

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Symbols:: $k$ : nonnegative integer, $n$ : nonnegative integer, $P$ : polynomial of two variables, $Q$ : polynomial of two variables, $x$ : real variable and $y$ : real variable
Proof sketch:: Use (18.20.5), (18.5.7), the symmetry of Jacobi polynomials given in Table 18.6.1, and (37.3.3).
Permalink:: http://dlmf.nist.gov/37.10.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §37.10(iv), §37.10 and Ch.37

…

8: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$

… ►

§37.3(i) Orthogonal Decomposition

… ►

37.3.1

W_{\alpha,\beta,\gamma}(x,y)=x^{\alpha}y^{\beta}(1-x-y)^{\gamma}

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Defines:: $W_{\alpha,\beta,\gamma}$ : weight function in two-variables on $\triangle$ (locally)
Symbols:: $\triangle$ : triangular region, $x$ : real variable, $y$ : real variable, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : real parameter ( $>-1$ )
Referenced by:: §37.5(iii)
Permalink:: http://dlmf.nist.gov/37.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(i), §37.3 and Ch.37

… ►

37.3.2

\langle{f},{g}\rangle_{\alpha,\beta,\gamma}=\frac{\Gamma\left(\alpha+\beta+% \gamma+3\right)}{\Gamma\left(\alpha+1\right)\Gamma\left(\beta+1\right)\Gamma% \left(\gamma+1\right)}\*\int_{\triangle}f(x,y)g(x,y)W_{\alpha,\beta,\gamma}(x,% y)\,\mathrm{d}x\,\mathrm{d}y,

\alpha,\beta,\gamma>-1

ⓘ

Defines:: $\langle{f},{g}\rangle_{\alpha,\beta,\gamma}$ : inner product on triangular region (locally), $\alpha$ : real parameter ( $>-1$ ) (locally), $\beta$ : real parameter ( $>-1$ ) (locally) and $\gamma$ : real parameter ( $>-1$ ) (locally)
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\triangle$ : triangular region, $x$ : real variable, $y$ : real variable, $W_{\alpha,\beta,\gamma}$ : weight function in two-variables on $\triangle$ , $f$ : two-variable function on $\triangle$ and $g$ : two-variable function on $\triangle$
Referenced by:: (37.3.18), (37.3.19), (37.3.20), §37.3(i)
Permalink:: http://dlmf.nist.gov/37.3.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(i), §37.3 and Ch.37

… ► … ►involving one-variable Jacobi polynomials

P^{(\alpha,\beta)}_{n}\left(x\right)

(see Table 18.3.1), are the case

w_{1}(x)=x^{\alpha}(1-x)^{\beta+\gamma}

w_{2}(x)=x^{\beta}(1-x)^{\gamma}

of (37.2.16). …

9: 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 $\left(1-x\right)^{\alpha}\left(x-y^{2}\right)^{\beta}$

… ►

37.7.2

W_{\alpha,\beta}(x,y)=(1-x)^{\alpha}(x-y^{2})^{\beta},

\alpha,\beta>-1

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Defines:: $W_{\alpha,\beta}$ : two-variable weight function on parabolic biangular region $\mathbb{A}$ (locally)
Symbols:: $x$ : real variable, $y$ : real variable and $\mathbb{A}$ : parabolic biangular region
Permalink:: http://dlmf.nist.gov/37.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §37.7 and Ch.37

… ►

37.7.19

yR_{k,n}^{\alpha,\frac{1}{2},\gamma}(1-x,y^{2})=\frac{(-1)^{n}{\left(1+k\right% )_{k+1}}}{{\left(\alpha+1+k\right)_{k+1}}}R_{2k+1,n+k+1}^{\alpha,\gamma}(x,y).

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $y$ : real variable and $R_{k,n}^{\alpha,\beta}$ : second two-variable Jacobi polynomial on the parabolic biangular region $\mathbb{A}$
Permalink:: http://dlmf.nist.gov/37.7.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §37.7(iv), §37.7 and Ch.37

… ►For the Laguerre and Hermite polynomials in one variable see Table 18.3.1. … ►

37.7.23

x{D}_{xx}+\tfrac{1}{2}{D}_{yy}+(\alpha+1-x){D}_{x}-y{D}_{y}.

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $x$ : real variable and $y$ : real variable
Referenced by:: item 5
Permalink:: http://dlmf.nist.gov/37.7.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §37.7(v), §37.7 and Ch.37

…

10: 1.4 Calculus of One Variable

§1.4 Calculus of One Variable

… ►

1.4.1

f(c+)\equiv\lim_{x\to c+}f(x)=f(c),

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Symbols:: $\equiv$ : equals by definition
Referenced by:: §1.4(ii)
Permalink:: http://dlmf.nist.gov/1.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §1.4(ii), §1.4 and Ch.1

… ►

§1.4(vi) Taylor’s Theorem for Real Variables

… ►

See accompanying text — Figure 1.4.2: Convex function $f(x)$ . … Magnify