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1: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$

… ►The monic basis

\{V^{\alpha,\beta,\gamma}_{k,n}\}_{0\leq k\leq n}

\mathcal{V}_{n}^{\alpha,\beta,\gamma}

and the co-monic basis

\{U^{\alpha,\beta,\gamma}_{k,n}\}_{0\leq k\leq n}

, biorthogonal to the monic basis, can be explicitly given as follows. … ►The first expression for

U^{\alpha,\beta,\gamma}_{k,n}

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

37.3.23

{D}_{x}V^{\alpha,\beta,\gamma}_{k,n}\left(x,y\right)=kV^{\alpha+1,\beta,\gamma% +1}_{k-1,n-1}\left(x,y\right),

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $V^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : monic basis element of two-variable Jacobi polynomial on the triangle, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $y$ : real variable, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : real parameter ( $>-1$ )
Proof sketch:: Use the first equality in (37.3.11).
Permalink:: http://dlmf.nist.gov/37.3.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(v), §37.3 and Ch.37

►

37.3.24

{D}_{x}\left(W_{\alpha+1,\beta,\gamma+1}(x,y)U^{\alpha+1,\beta,\gamma+1}_{k-1,% n-1}\left(x,y\right)\right)=W_{\alpha,\beta,\gamma}(x,y)U^{\alpha,\beta,\gamma% }_{k,n}\left(x,y\right),

ⓘ

Symbols:: $U^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : co-monic basis element of two-variable Jacobi polynomial on the triangle, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $y$ : real variable, $W_{\alpha,\beta,\gamma}$ : weight function in two-variables on $\triangle$ , $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : real parameter ( $>-1$ )
Proof sketch:: Use the first equality in (37.3.12).
Permalink:: http://dlmf.nist.gov/37.3.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(v), §37.3 and Ch.37

►

37.3.25

{D}_{y}V^{\alpha,\beta,\gamma}_{k,n}\left(x,y\right)=(n-k)V^{\alpha,\beta+1,% \gamma+1}_{k,n-1}\left(x,y\right),

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $V^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : monic basis element of two-variable Jacobi polynomial on the triangle, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $y$ : real variable, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : real parameter ( $>-1$ )
Proof sketch:: Use the first equality in (37.3.11).
Permalink:: http://dlmf.nist.gov/37.3.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(v), §37.3 and Ch.37

…

2: 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\leq k\leq n}

\mathcal{V}_{n}^{\alpha}

and the co-monic basis

\{U_{k,n}^{(\alpha+\frac{1}{2})}\}_{0\leq k\leq n}

, biorthogonal to the monic basis, can be explicitly given as follows. … ►

37.4.40

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

ⓘ

Symbols:: $V^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : monic basis element of two-variable Jacobi polynomial on the triangle, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $y$ : real variable and $V_{k,n}^{(\alpha+\frac{1}{2})}$ : monic basis element on the disk
Permalink:: http://dlmf.nist.gov/37.4.E40
Encodings:: pMML, png, TeX
See also:: Annotations for §37.4(vii), §37.4 and Ch.37

►

37.4.41

yV^{-\frac{1}{2},\frac{1}{2},\gamma}_{k,n}\left(x^{2},y^{2}\right)=V_{2k,2n+1}% ^{(\gamma+\frac{1}{2})}(x,y),

ⓘ

Symbols:: $V^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : monic basis element of two-variable Jacobi polynomial on the triangle, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $y$ : real variable and $V_{k,n}^{(\alpha+\frac{1}{2})}$ : monic basis element on the disk
Permalink:: http://dlmf.nist.gov/37.4.E41
Encodings:: pMML, png, TeX
See also:: Annotations for §37.4(vii), §37.4 and Ch.37

►

37.4.42

xV^{\frac{1}{2},-\frac{1}{2},\gamma}_{k,n}\left(x^{2},y^{2}\right)=V_{2k+1,2n+% 1}^{(\gamma+\frac{1}{2})}(x,y),

ⓘ

Symbols:: $V^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : monic basis element of two-variable Jacobi polynomial on the triangle, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $y$ : real variable and $V_{k,n}^{(\alpha+\frac{1}{2})}$ : monic basis element on the disk
Permalink:: http://dlmf.nist.gov/37.4.E42
Encodings:: pMML, png, TeX
See also:: Annotations for §37.4(vii), §37.4 and Ch.37

…

3: 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.

4: 37.5 Quarter Plane with Weight Function $x^{\alpha}y^{\beta}{\mathrm{e}}^{-x-y}$

… ►

37.5.14

\lim_{\gamma\to\infty}\gamma^{n}V^{\alpha,\beta,\gamma}_{k,n}\left(\gamma^{-1}% x,\gamma^{-1}y\right)=(-1)^{n}k!\,(n-k)!\,L^{(\alpha)}_{k}\left(x\right)L^{(% \beta)}_{n-k}\left(y\right),

ⓘ

Symbols:: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $!$ : factorial (as in $n!$ ), $V^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : monic basis element of two-variable Jacobi polynomial on the triangle, $k$ : nonnegative integer, $n$ : 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
Encodings:: pMML, png, TeX
See also:: Annotations for §37.5(iii), §37.5 and Ch.37

►

37.5.15

\lim_{\gamma\to\infty}U^{\alpha,\beta,\gamma}_{k,n}\left(\gamma^{-1}x,\gamma^{% -1}y\right)=k!\,(n-k)!\,L^{(\alpha)}_{k}\left(x\right)L^{(\beta)}_{n-k}\left(y% \right).

ⓘ

Symbols:: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $U^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : co-monic basis element of two-variable Jacobi polynomial on the triangle, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $n$ : 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
Encodings:: pMML, png, TeX
See also:: Annotations for §37.5(iii), §37.5 and Ch.37

5: 18.4 Graphics

… ►

See accompanying text — Figure 18.4.7: Monic Hermite polynomials $h_{n}(x)=2^{-n}H_{n}\left(x\right)$ , $n=1,2,3,4,5$ . Magnify

…

6: 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

…

7: 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. …

8: 18.2 General Orthogonal Polynomials

… ►(ii) monic OP’s:

k_{n}=1

. … ►

Monic and Orthonormal Forms

… ►the monic recurrence relations (18.2.8) and (18.2.10) take the form … ►Define the first associated monic orthogonal polynomials

p_{n}^{(1)}(x)

as monic OP’s satisfying …

9: 3.5 Quadrature

… ►

Gauss–Legendre Formula

… ►

Gauss–Jacobi Formula

… ►

Gauss–Laguerre Formula

… ►

Gauss–Hermite Formula

… ►All the monic orthogonal polynomials

\{p_{n}\}

used with Gauss quadrature satisfy a three-term recurrence relation (§18.2(iv)): …

10: 37.13 General Orthogonal Polynomials of $d$ Variables

… ►The monic basis of

\mathcal{V}_{n}^{d}

consists of polynomials

P_{\boldsymbol{{\nu}}}

(

|{\boldsymbol{{\nu}}}|=n

) such that

P_{\boldsymbol{{\nu}}}(\mathbf{x})=\mathbf{x}^{\boldsymbol{{\nu}}}+\mbox{% polynomial of degree less than $n$}

. …In the co-monic basis

\{Q_{\boldsymbol{{\nu}}}\}_{|{\boldsymbol{{\nu}}}|=n}

, biorthogonal to the monic basis,

Q_{\boldsymbol{{\nu}}}

is a polynomial of degree

n

which is orthogonal to

\mathbf{x}^{\boldsymbol{{\mu}}}

(

|{\boldsymbol{{\mu}}}|\leq n

\boldsymbol{{\mu}}\neq\boldsymbol{{\nu}}

) with respect to the inner product (37.13.1), analogous to (37.2.4). …