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

… ►The monic basis ${\{V_{k,n}^{(\alpha +\frac{1}{2})}\}}_{0\le k\le n}$ of ${𝒱}_n^{\alpha }$ and the co-monic basis ${\{U_{k,n}^{(\alpha +\frac{1}{2})}\}}_{0\le k\le n}$, biorthogonal to the monic basis, can be explicitly given as follows. … ►

37.4.40 $$V_{k,n}^{-\frac{1}{2},-\frac{1}{2},\gamma }(x^2,y^2)=V_{2k,2n}^{(\gamma +\frac{1}{2})}(x,y),$$

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Symbols:

: monic basis element of two-variable Jacobi polynomial on the triangle, $V_{k,n}^{((\alpha +\frac{1}{2}))}(x,y)$: monic basis element disk $V$ , $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable and $\gamma$: real parameter

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http://dlmf.nist.gov/37.4.E40

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Annotations for §37.4(vii), §37.4 and Ch.37

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37.4.41 $$y{V}_{k,n}^{-\frac{1}{2},\frac{1}{2},\gamma }(x^2,y^2)=V_{2k,2n+1}^{(\gamma +\frac{1}{2})}(x,y),$$

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Symbols:

: monic basis element of two-variable Jacobi polynomial on the triangle, $V_{k,n}^{((\alpha +\frac{1}{2}))}(x,y)$: monic basis element disk $V$ , $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable and $\gamma$: real parameter

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http://dlmf.nist.gov/37.4.E41

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Annotations for §37.4(vii), §37.4 and Ch.37

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37.4.42 $$x{V}_{k,n}^{\frac{1}{2},-\frac{1}{2},\gamma }(x^2,y^2)=V_{2k+1,2n+1}^{(\gamma +\frac{1}{2})}(x,y),$$

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Symbols:

: monic basis element of two-variable Jacobi polynomial on the triangle, $V_{k,n}^{((\alpha +\frac{1}{2}))}(x,y)$: monic basis element disk $V$ , $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable and $\gamma$: real parameter

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http://dlmf.nist.gov/37.4.E42

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Annotations for §37.4(vii), §37.4 and Ch.37

…

2: 37.15 Orthogonal Polynomials on the Ball

… ►The monic basis ${\{V_{𝝂}^{(\alpha +\frac{1}{2})}\}}_{|𝝂|=n}$ of ${𝒱}_n^{\alpha }\left({𝔹}^d\right)$ and the co-monic basis ${\{U_{𝝂}^{(\alpha +\frac{1}{2})}\}}_{|𝝂|=n}$, biorthogonal to the monic basis, can be explicitly given by … ►

37.15.12 $${⟨V_{𝝂}^{(\alpha +\frac{1}{2})},U_{𝝁}^{(\alpha +\frac{1}{2})}⟩}_{\alpha }={(-1)}^{|𝝂|}𝝂!\frac{{\left(\alpha +1\right)}_{|𝝂|}}{{\left(\alpha +\frac{1}{2}d+1\right)}_{|𝝂|}}{\delta }_{𝝂,𝝁}.$$

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Symbols:

${\delta }_{𝝂,𝝁}$: Kronecker delta for multi-indices, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $U_{𝝂}^{(𝜶)}\left(𝐱\right)$: co-monic basis element on the ball, $!$: factorial for multi-indices, $V_{𝝂}^{(𝜶)}\left(𝐱\right)$: monic basis element on the ball, $d$: positive integer, usually $\ge 2$ , $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions, $𝝁$: multi-index of $d$-dimensions, $\alpha$: real variable and ${⟨f,g⟩}_{\alpha }$: inner product on $d$-dimensional ball

Proved:

Dunkl and Xu (2014, Proposition 5.2.6)(proved)

Permalink:

http://dlmf.nist.gov/37.15.E12

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

… ►

37.15.14 $$C_n^{(\alpha +\frac{1}{2}d)}(⟨𝐱,𝐲⟩)=\sum _{|𝝂|=n}\frac{2^n{\left(\alpha +\frac{1}{2}d\right)}_n}{𝝂!}{𝐲}^{𝝂}{V}_{𝝂}^{(\alpha +\frac{1}{2})}\left(𝐱\right),$$ $𝐲\in {𝕊}^{d-1}$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\in$: element of, $!$: factorial for multi-indices, $⟨𝐮,𝐯⟩$: inner product over vectors, $V_{𝝂}^{(𝜶)}\left(𝐱\right)$: monic basis element on the ball, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space, $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions, ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ and $\alpha$: real variable

Proof sketch:

Use (18.12.4) and (37.15.13).

Permalink:

http://dlmf.nist.gov/37.15.E14

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

… ►

37.15.23 $${𝐱}^{𝐤}{V}_{-\frac{\mathrm{𝟏}}{\mathrm{𝟐}}+𝐤,\alpha }^{𝝂}\left({𝐱}^2\right)=\mathrm{const}.V_{2𝝂+𝐤}^{(\alpha +\frac{1}{2})}\left(𝐱\right),$$ $𝐤\in {\{0,1\}}^d$,

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Symbols:

$\in$: element of, $V_{𝝂}^{(𝜶)}\left(𝐱\right)$: monic basis element on the ball, $V_{𝝂}^{𝜶}\left(𝐱\right)$: monic basis element on the simplex, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝝂$: multi-index of $d$-dimensions, $𝐤$: element of ${\{0,1\}}^d$ and $\alpha$: real variable

Permalink:

http://dlmf.nist.gov/37.15.E23

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Annotations for §37.15(vii), §37.15 and Ch.37

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37.15.24 $${𝐱}^{𝐤}{U}_{𝝂}^{-\frac{\mathrm{𝟏}}{\mathrm{𝟐}}+𝐤,\alpha }\left({𝐱}^2\right)=\mathrm{const}.U_{2𝝂+𝐤}^{(\alpha +\frac{1}{2})}\left(𝐱\right),$$ $𝐤\in {\{0,1\}}^d$.

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Symbols:

$U_{𝝂}^{(𝜶)}\left(𝐱\right)$: co-monic basis element on the ball, $U_{𝝂}^{𝜶}\left(𝐱\right)$: co-monic basis element on the simplex, $\in$: element of, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝝂$: multi-index of $d$-dimensions, $𝐤$: element of ${\{0,1\}}^d$ and $\alpha$: real variable

Permalink:

http://dlmf.nist.gov/37.15.E24

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See also:

Annotations for §37.15(vii), §37.15 and Ch.37

3: 37.3 Triangular Region with Weight Function

… ►The monic basis of and the co-monic basis , biorthogonal to the monic basis, can be explicitly given as follows. … ►The first expression for in (37.3.12) is an analogue of the Rodrigues formulas in §18.5(ii). … ►

37.3.23

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , : monic basis element of two-variable Jacobi polynomial on the triangle, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proof sketch:

Use the first equality in (37.3.11).

Permalink:

http://dlmf.nist.gov/37.3.E23

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

►

37.3.24

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Symbols:

: co-monic basis element of two-variable Jacobi polynomial on the triangle, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, : weight function in two-variables on $\mathrm{△}$ , : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proof sketch:

Use the first equality in (37.3.12).

Permalink:

http://dlmf.nist.gov/37.3.E24

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

►

37.3.25

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , : monic basis element of two-variable Jacobi polynomial on the triangle, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proof sketch:

Use the first equality in (37.3.11).

Permalink:

http://dlmf.nist.gov/37.3.E25

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

…

4: 29.21 Tables

… ►

Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^2=0.1(.1)0.9$, $n=1(1)30$. Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

5: 37.14 Orthogonal Polynomials on the Simplex

… ►The monic basis ${\{V_{𝝂}^{𝜶}\}}_{|𝝂|=n}$ of ${𝒱}_n^{𝜶}\left({\mathrm{△}}^d\right)$ and the co-monic basis ${\{U_{𝝂}^{𝜶}\}}_{|𝝂|=n}$, biorthogonal to the monic basis, can be explicitly given by … ►

37.14.9 $$U_{𝝂}^{𝜶}\left(𝐱\right)=W_{𝜶}{(𝐱)}^{-1}{D}_{𝐱}^{𝝂}\left({𝐱}^{𝝂}{(1-|𝐱|)}^{|𝝂|}{W}_{𝜶}(𝐱)\right),$$ $𝝂\in {\mathbb{N}}_0^d$, $|𝝂|=n$.

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Defines:

$U_{𝝂}^{𝜶}\left(𝐱\right)$: co-monic basis element on the simplex

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, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions, ${\mathbb{N}}_0$: set of all nonnegative integers, $W_{𝜶}$: weight function on $d$-variables on the simplex and : real parameter ($>-1$)

Source:

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

Referenced by:

§37.14(iii), §37.15(vii)

Permalink:

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

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

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

37.14.10

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Symbols:

${\delta }_{𝝂,𝝁}$: Kronecker delta for multi-indices, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $U_{𝝂}^{𝜶}\left(𝐱\right)$: co-monic basis element on the simplex, $!$: factorial (as in $n!$), $V_{𝝂}^{𝜶}\left(𝐱\right)$: monic basis element on the simplex, $d$: positive integer, usually $\ge 2$ , $𝟏$: multi-dimensional vector with all components being unity, $\mathrm{\ell }$: positive integer, $𝜶$: vector $=({\alpha }_1,\mathrm{\dots },{\alpha }_{d+1})$ , $|𝐱|$: $l1$ norm, $𝝂$: multi-index of $d$-dimensions, $𝝁$: multi-index of $d$-dimensions, : real parameter ($>-1$) and ${⟨f,g⟩}_{𝜶}$: inner product on $d$-dimensional simplex

Proved:

Dunkl and Xu (2014, Proposition 5.3.3); There replace by ($\mathrm{\ell }=1,\mathrm{\dots },d+1$). Also note an error in the formula there. Before the integral on the left a factor $w_{𝜶}^T$ should be inserted.(proved)

Permalink:

http://dlmf.nist.gov/37.14.E10

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

►See Littler and Fackerell (1975, (2.7)) for an expression of $V_{𝝂}^{𝜶}\left(𝐱\right)$ in terms of Lauricella’s hypergeometric function $F_A$. …

6: 37.5 Quarter Plane with Weight Function $x^{\alpha }{y}^{\beta }{\mathrm{e}}^{-x-y}$

… ►

37.5.14 $$\underset{\gamma \to \mathrm{\infty }}{lim}{\gamma }^n{V}_{k,n}^{\alpha ,\beta ,\gamma }({\gamma }^{-1}x,{\gamma }^{-1}y)={(-1)}^{n}k!(n-k)!L_k^{(\alpha )}\left(x\right)L_{n-k}^{(\beta )}\left(y\right),$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), : monic basis element of two-variable Jacobi polynomial on the triangle, $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $\gamma$: real parameter and $\beta$: real parameter

Permalink:

http://dlmf.nist.gov/37.5.E14

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See also:

Annotations for §37.5(iii), §37.5 and Ch.37

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37.5.15 $$\underset{\gamma \to \mathrm{\infty }}{lim}{U}_{k,n}^{\alpha ,\beta ,\gamma }({\gamma }^{-1}x,{\gamma }^{-1}y)=k!(n-k)!L_k^{(\alpha )}\left(x\right)L_{n-k}^{(\beta )}\left(y\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, : co-monic basis element of two-variable Jacobi polynomial on the triangle, $!$: factorial (as in $n!$), $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable, $y$: real variable, $\alpha$: real parameter, $\gamma$: real parameter and $\beta$: real parameter

Permalink:

http://dlmf.nist.gov/37.5.E15

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Annotations for §37.5(iii), §37.5 and Ch.37

7: 37.6 Plane with Weight Function ${\mathrm{e}}^{-x^2-y^2}$

… ►

37.6.17 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{\frac{1}{2}n}{V}_{k,n}^{(\alpha +\frac{1}{2})}({\alpha }^{-\frac{1}{2}}x,{\alpha }^{-\frac{1}{2}}y)=2^{-n}{H}_k\left(x\right)H_{n-k}\left(y\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $V_{k,n}^{((\alpha +\frac{1}{2}))}(x,y)$: monic basis element disk $V$ , $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/37.6.E17

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Annotations for §37.6(iv), §37.6 and Ch.37

►

37.6.18 $$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{-\frac{1}{2}n}{U}_{k,n}^{(\alpha +\frac{1}{2})}({\alpha }^{-\frac{1}{2}}x,{\alpha }^{-\frac{1}{2}}y)={(-1)}^n{H}_k\left(x\right)H_{n-k}\left(y\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $U_{k,n}^{((\alpha +\frac{1}{2}))}(x,y)$: co-monic basis element disk $U$ , $n$: nonnegative integer, $k$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/37.6.E18

Encodings:

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See also:

Annotations for §37.6(iv), §37.6 and Ch.37

8: 18.4 Graphics

… ►

►►

Figure 18.4.7: Monic Hermite polynomials $h_n(x)=2^{-n}{H}_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

…

9: 18.30 Associated OP’s

… ►The lowest order monic versions of both of these appear in §18.2(x), (18.2.31) defining the $c=1$ associated monic polynomials, and (18.2.32) their closely related cousins the $c=0$ corecursive polynomials. … ►

§18.30(vii) Corecursive and Associated Monic Orthogonal Polynomials

… ►

Associated Monic OP’s

… ►

The “Zeroth” Corecursive Monic OP

… ►

Relationship of Monic Corecursive and Monic Associated OP’s

…

10: 37.20 Mathematical Applications

… ►The $L^2$ norms of the monic OPs are the error of the least square approximation of monomials by polynomials of lower degrees. …