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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: 37.16 Orthogonal Polynomials on the Hyperoctant

§37.16 Orthogonal Polynomials on the Hyperoctant

… ►

37.16.2 $$W_{𝜶}(𝐱)={𝐱}^{𝜶}{\mathrm{e}}^{-|𝐱|},$$ ${\alpha }_1,\mathrm{\dots },{\alpha }_d>-1$.

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Defines:

$W$: weight function in $d$-variables on the hyperoctant (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ and $|𝐱|$: $l1$ norm

Permalink:

http://dlmf.nist.gov/37.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.16 and Ch.37

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37.16.3 $${⟨f,g⟩}_{𝜶}=\frac{1}{{\prod }_{\mathrm{\ell }=1}^d\mathrm{\Gamma }\left({\alpha }_{\mathrm{\ell }}+1\right)}{\int }_{{ℝ}_{+}^d}f(𝐱)g(𝐱)W_{𝜶}(𝐱)d𝐱,$$ ${\alpha }_1,\mathrm{\dots },{\alpha }_d>-1$,

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Defines:

${⟨f,g⟩}_{𝜶}$: inner product on hyperoctant (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $ℝ$: real line, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , ${ℝ}_{+}^d$: hyperoctant and $W$: weight function in $d$-variables on the hyperoctant

Referenced by:

§37.16

Permalink:

http://dlmf.nist.gov/37.16.E3

Encodings:

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See also:

Annotations for §37.16 and Ch.37

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37.16.4 $$\sum _{\mathrm{\ell }=1}^d\left(x_{\mathrm{\ell }}{D}_{x_{\mathrm{\ell }}}^2+({\alpha }_{\mathrm{\ell }}+1-x_{\mathrm{\ell }})D_{x_{\mathrm{\ell }}}\right)u(x)=-nu(x),$$ $u\in {𝒱}_n^{𝜶}\left({ℝ}_{+}^d\right)$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $ℝ$: real line, ${𝒱}_n^{\alpha }\left({ℝ}_{+}^d\right)$: vector space of orthogonal polynomials on the hyperoctant, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , ${ℝ}_{+}^d$: hyperoctant and $x$: real variable

Proved:

Dunkl and Xu (2014, (5.1.9))(proved)

Referenced by:

§37.13(ii)

Permalink:

http://dlmf.nist.gov/37.16.E4

Encodings:

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See also:

Annotations for §37.16 and Ch.37

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37.16.7 $${𝐏}_z^{𝜶}(𝐱,𝐲)=\sum _{n=0}^{\mathrm{\infty }}{𝐑}_n^{𝜶}(𝐱,𝐲)z^n={(1-z)}^{-1}\exp \left(\frac{z(|𝐱|+|𝐲|)}{z-1}\right)\prod _{\mathrm{\ell }=1}^d\frac{\mathrm{\Gamma }\left({\alpha }_{\mathrm{\ell }}+1\right)}{{\left(x_{\mathrm{\ell }}{y}_{\mathrm{\ell }}z\right)}^{\frac{1}{2}{\alpha }_{\mathrm{\ell }}}}{I}_{{\alpha }_{\mathrm{\ell }}}\left(\frac{2\sqrt{x_{\mathrm{\ell }}{y}_{\mathrm{\ell }}z}}{1-z}\right),$$ , $𝐱,𝐲\in {ℝ}_{+}^d$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $|z|$: absolute value (modulus) for complex numbers, $\in$: element of, $\exp z$: exponential function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $ℝ$: real line, $n$: nonnegative integer, $z$: complex number, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $|𝐱|$: $l1$ norm, ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ , ${𝐏}_z$: Poisson kernel on ${𝒱}_n^d$ , ${ℝ}_{+}^d$: hyperoctant, $x$: real variable and $y$: real variable

Proved:

Dunkl and Xu (2014, Proposition 8.4.8); There set $V_{\kappa }=\mathrm{id}$.(proved)

Permalink:

http://dlmf.nist.gov/37.16.E7

Encodings:

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See also:

Annotations for §37.16(ii), §37.16 and Ch.37

…

4: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►Let … ►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 $x_{\mathrm{\ell }}$ variables. … ►In one variable they are essentially ultraspherical, Jacobi, continuous -ultraspherical, or Askey–Wilson polynomials. …

5: 37.10 Other Orthogonal Polynomials of Two Variables

… ►

§37.10(ii) Orthogonal Polynomials on an Annulus

►The Tatian polynomials (see Tatian (1974)) are OPs of the form (37.2.27) on the circular region () with weight function 1. Thus the $p_n^{(k)}(x)$ in (37.2.27) are orthogonal on $(\rho ,1)$ with weight function $x^k$. … ►For any $k=0,\mathrm{\dots },m$ let $h_k(y)$ be polynomials of $y$ with real coefficients of degree at most $\frac{m}{2}-\left|\frac{m}{2}-k\right|$, with $h_0(y)=1$, such that for all $-1\le y\le 1$, … ►See §18.31 for Bernstein–Szegő polynomials of one variable. …

6: 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. …

7: Bibliography I

… ►

Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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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

►

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

…

8: 37.13 General Orthogonal Polynomials of $d$ Variables

… ►Let $W$ be a nonnegative weight function on an open set $\mathrm{\Omega }$ in ${ℝ}^d$ such that the integral ${\int }_{\mathrm{\Omega }}P(𝐱)W(𝐱)d𝐱$ is well-defined and absolutely convergent for all polynomials $P$, and such that ${\int }_{\mathrm{\Omega }}W(𝐱)d𝐱>0$. …on the space of polynomials of $d$ variables with real coefficients. … ►

37.13.4 $${\int }_{\mathrm{\Omega }}{𝐑}_n(𝐱,𝐲)Q(𝐲)W(𝐲)d𝐲=Q(𝐱),$$ $Q\in {𝒱}_n^d$, $𝐱\in {ℝ}^d$.

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Defines:

$Q$: polynomial of $d$-variables (locally) and ${𝐑}_n$: reproducing kernel on ${𝒱}_n^d$ (locally)

Symbols:

$dx$: differential of $x$ , $\in$: element of, $\int$: integral, $ℝ$: real line, ${𝒱}_n^d$: vector space of orthogonal orthogonal polynomials in $d$-variables, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${ℝ}^d$: $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $\mathrm{\Omega }$: open set in $d$-dimensional Euclidean space, $W$: nonnegative weight function in $d$-variables on open set $\mathrm{\Omega }$ and $𝐱$: $d$-dimensional vector of real numbers

Permalink:

http://dlmf.nist.gov/37.13.E4

Encodings:

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See also:

Annotations for §37.13, §37.13 and Ch.37

… ►Analogous to the Krall–Sheffer classification in the two-variable case (see §37.2(viii)), one can ask for admissible $d$-variable second order PDOs $L$, i. … ►The corresponding OPs are products of one-variable OPs of Hahn class, as defined in §18.19. …

9: 37.14 Orthogonal Polynomials on the Simplex

… ►

§37.14(i) Orthogonal Decomposition

… ►

37.14.2 , , $𝐱\in {\mathrm{△}}^d$,

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Defines:

$W_{𝜶}$: weight function on $d$-variables on the simplex (locally) and : real parameter ($>-1$) (locally)

Symbols:

$\in$: element of, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $|𝐱|$: $l1$ norm, ${\mathrm{△}}^d$: simplex and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.14.E2

Encodings:

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See also:

Annotations for §37.14(i), §37.14 and Ch.37

… ►For the Jacobi polynomial $P_n^{(\alpha ,\beta )}\left(x\right)$ of one variable see Table 18.3.1. … ►The spaces ${𝒱}_n^{𝜶}\left({\mathrm{△}}^d\right)$ are eigenspaces of a second order partial differential operator: … ►

37.14.12 $u\in {𝒱}_n^{𝜶}\left({\mathrm{△}}^d\right)$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, ${𝒱}_n^{𝜶}\left({\mathrm{△}}^d\right)$: vector space of orthogonal polynomials on the simplex, $n$: nonnegative integer, $j$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝟏$: multi-dimensional vector with all components being unity, $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $|𝐱|$: $l1$ norm, ${\mathrm{△}}^d$: simplex, $W_{𝜶}$: weight function on $d$-variables on the simplex, $u$: polynomial in $d$-variables on the simplex and $x$: real variable

Permalink:

http://dlmf.nist.gov/37.14.E12

Encodings:

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See also:

Annotations for §37.14(iv), §37.14 and Ch.37

…

10: 37.3 Triangular Region with Weight Function

… ►

§37.3(i) Orthogonal Decomposition

… ►

37.3.1

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Defines:

: weight function in two-variables on $\mathrm{△}$ (locally)

Symbols:

$\mathrm{△}$: triangular region, $x$: real variable, $y$: real variable, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Referenced by:

§37.5(iii)

Permalink:

http://dlmf.nist.gov/37.3.E1

Encodings:

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See also:

Annotations for §37.3(i), §37.3 and Ch.37

… ► … ►involving one-variable Jacobi polynomials (see Table 18.3.1), are the case , of (37.2.16). … ►

37.3.15 ,

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, : vector space of two-variable orthogonal polynomials on the triangular region, $n$: nonnegative integer, $z$: complex number, $x$: real variable, $y$: real variable, : weight function in two-variables on $\mathrm{△}$ , $u$: OP in two variables on the triangular region, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proof sketch:

By direct computation the left-hand side of the present equality equals the left-hand side of (37.3.14).

Permalink:

http://dlmf.nist.gov/37.3.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §37.3(iv), §37.3 and Ch.37

…