with 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: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

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Hahn $\to$ Jacobi

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Meixner $\to$ Laguerre

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Charlier $\to$ Hermite

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Figure 18.21.1: Askey scheme. … Magnify

3: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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Legendre, Ultraspherical, and Jacobi

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Jacobi $\to$ Laguerre

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Laguerre $\to$ Hermite

… ► See §18.11(ii) for limit formulas of Mehler–Heine type.

4: 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(ii) Limit Relations

… ►See also Figure 18.21.1. … ►

§18.26(iv) Generating Functions

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5: 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 $𝕡$

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§37.7(iv) A Second System of OPs

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6: 37.10 Other Orthogonal Polynomials of Two Variables

§37.10 Other Orthogonal Polynomials of Two Variables

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7: 18.11 Relations to Other Functions

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Ultraspherical

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8: 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, Part 37.III and Ch.37

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9: 37.19 Other Orthogonal Polynomials of $d$ Variables

§37.19 Other Orthogonal Polynomials of $d$ Variables

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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 …