relations to other orthogonal polynomials

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1: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

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

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§37.7(i) Jacobi polynomials on the parabolic biangular region $𝔸$

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§37.7(iv) A Second System of OPs

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3: 18.20 Hahn Class: Explicit Representations

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§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

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4: 37.16 Orthogonal Polynomials on the Hyperoctant

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37.16.5 $$\prod _{\mathrm{\ell }=1}^d{L}_{{\nu }_{\mathrm{\ell }}}^{({\alpha }_{\mathrm{\ell }})}\left(x_{\mathrm{\ell }}\right),$$ ${\nu }_1+\mathrm{\cdots }+{\nu }_d=n$.

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer and $x$: real variable

Proved:

Dunkl and Xu (2014, (5.1.8))(proved)

Referenced by:

§37.16(iii)

Permalink:

http://dlmf.nist.gov/37.16.E5

Encodings:

pMML, png, TeX

See also:

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

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5: 18.5 Explicit Representations

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§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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6: 18.38 Mathematical Applications

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7: 37.18 Orthogonal Polynomials on Quadratic Domains

… ►The OPs in the basis (37.18.5) are given in terms of the Jacobi polynomials and OPs in ${𝒱}_m({𝔹}^d)$ with respect to $W_{\mu -\frac{1}{2}}$: … ►

37.18.13 $$Q_{𝐤,m}^n(𝐱,t;\mu ,\beta )=L_{n-m}^{(2m+2\mu +\beta +d-1)}\left(t\right)t^m{P}_{𝐤}^m\left(\frac{𝐱}{t}\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer, $m$: nonnegative integer, $P$: polynomial of two variables, $Q$: polynomial of two variables, $d$: positive integer, usually $\ge 2$ and $𝐱$: vector in $d$-dimensional Euclidean space

Referenced by:

§37.18(iii)

Permalink:

http://dlmf.nist.gov/37.18.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.18(iii), §37.18 and Ch.37

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8: 18.26 Wilson Class: Continued

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§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

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§18.26(iv) Generating Functions

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9: 15.9 Relations to Other Functions

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§15.9(i) Orthogonal Polynomials

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15.9.10 $$P_n^{(\lambda )}(x;\varphi )=\frac{{\left(2\lambda \right)}_n}{n!}{\mathrm{e}}^{n\mathrm{i}\varphi }F(\genfrac{}{}{0pt}{}{-n,\lambda +\mathrm{i}x}{2\lambda };1-{\mathrm{e}}^{-2\mathrm{i}\varphi }).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

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10: 37.5 Quarter Plane with Weight Function $x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y}$

… ►Obviously, an orthogonal basis of ${𝒱}_n^{\alpha ,\beta }$ consisting of products of Laguerre polynomials is given by … ►Define Laguerre–Jacobi polynomials on the quarter plane by …