Rodrigues formulas

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1: 18.20 Hahn Class: Explicit Representations

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§18.20(i) Rodrigues Formulas

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Table 18.20.1: Krawtchouk, Meixner, and Charlier OP’s: Rodrigues formulas (18.20.1).

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$p_n(x)$	$F(x)$	${\kappa }_n$
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2: 18.5 Explicit Representations

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§18.5(ii) Rodrigues Formulas

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Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).

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$p_n(x)$	$w(x)$	$F(x)$	${\kappa }_n$
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3: 18.3 Definitions

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As given by a Rodrigues formula (18.5.5).

… ►For representations of the polynomials in Table 18.3.1 by Rodrigues formulas, see §18.5(ii). …

4: 37.14 Orthogonal Polynomials on the Simplex

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37.14.9 $𝝂\in {\mathbb{N}}_0^d$, $|𝝂|=n$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $\mathbb{N}$: set of all positive integers, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space, ${\mathbb{N}}_0$: set of all nonnegative integers and ${\alpha }_{\mathrm{\ell }}$: real parameter ($>-1$)

Source:

Dunkl and Xu (2014, §5.3); See under the heading “Rodrigues formula and biorthogonal basis”. In the formula there replace ${\alpha }_{\mathrm{\ell }}$ by ${\alpha }_{\mathrm{\ell }}+\frac{1}{2}$ ($\mathrm{\ell }=1,\mathrm{\dots },d+1$).

Referenced by:

§37.14(iii), §37.15(vii)

Permalink:

http://dlmf.nist.gov/37.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §37.14(iii), §37.14, Part 37.III and Ch.37

►Formula (37.14.9) is an analogue of the Rodrigues formulas in §18.5(ii). …

5: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

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Rodrigues Type Formula

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6: 14.7 Integer Degree and Order

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§14.7(ii) Rodrigues-Type Formulas

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7: 37.4 Disk with Weight Function ${\left(1-x^2-y^2\right)}^{\alpha }$

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Rodrigues type formula

… ►The first equality in (37.4.24) is an analogue of the Rodrigues formulas in §18.5(ii). Clearly, …

8: 37.15 Orthogonal Polynomials on the Ball

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37.15.11 $$U_{𝝂}^{(\alpha +\frac{1}{2})}(𝐱)=W_{\alpha }{(𝐱)}^{-1}{D}_{𝐱}^{𝝂}\left({(1-{\Vert 𝐱\Vert }^2)}^{|𝝂|}{W}_{\alpha }(𝐱)\right)={\left(\alpha +1\right)}_{|𝝂|}\sum _{𝝁\le \frac{1}{2}𝝂}\frac{{\left(-𝝂\right)}_{2𝝁}}{{\left(\alpha +1\right)}_{|𝝁|}|𝝁|!}{(-2𝐱)}^{𝝂-2𝝁}{({\Vert 𝐱\Vert }^2-1)}^{|𝝁|},$$ $𝝂\in {\mathbb{N}}_0^d$, $|𝝂|=n$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\in$: element of, $!$: factorial (as in $n!$), $\mathbb{N}$: set of all positive integers, $\Vert 𝐯\Vert$: vector norm ($l$²), $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, ${\mathbb{N}}_0$: set of all nonnegative integers and $W$: weight function in $d$-variables

Proved:

Dunkl and Xu (2014, Propositions 5.2.5 and 5.2.7); There the polynomial $U_{𝝂}^{(\alpha +\frac{1}{2})}$ is denoted by $U_{\alpha }$. The formulas match up to constant factors.(proved)

Referenced by:

§37.15(iv), §37.15(vii)

Permalink:

http://dlmf.nist.gov/37.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(iv), §37.15, Part 37.III and Ch.37

►The first equality in (37.15.11) is an analogue of the Rodrigues formulas in §18.5(ii). …

9: 18.10 Integral Representations

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10: 37.3 Triangular Region with Weight Function

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37.3.12

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Defines:

: co-monic basis element of two-variable Jacobi polynomial on the triangular region (locally)

Symbols:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function, $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $(a,b)$: open interval, $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $x$: real variable, $y$: real variable, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proveds:

Appell and Kampé de Fériet (1926, Chap. VI); Dunkl and Xu (2014, Proposition 5.3.3); for $d=2$(proved)

Source:

Erdélyi et al. (1953b, 12.4(4))

Referenced by:

(37.3.24), (37.3.26), §37.3(iii), §37.3(v), §37.4(vii), §37.5(iii)

Permalink:

http://dlmf.nist.gov/37.3.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §37.3(iii), §37.3, Part 37.II and Ch.37

… ►The first expression for in (37.3.12) is an analogue of the Rodrigues formulas in §18.5(ii). …