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1: 29.21 Tables

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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.

2: 18.4 Graphics

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

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3: 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

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Associated Monic OP’s

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The “Zeroth” Corecursive Monic OP

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Relationship of Monic Corecursive and Monic Associated OP’s

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4: 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. …

5: 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 …

6: 3.5 Quadrature

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Gauss–Legendre Formula

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Gauss–Jacobi Formula

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Gauss–Laguerre Formula

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Gauss–Hermite Formula

… ►All the monic orthogonal polynomials $\{p_n\}$ used with Gauss quadrature satisfy a three-term recurrence relation (§18.2(iv)): …

7: 37.13 General Orthogonal Polynomials of $d$ Variables

… ►The monic basis of ${𝒱}_n^d$ consists of polynomials $P_{𝝂}$ ($|𝝂|=n$) such that $P_{𝝂}(𝐱)={𝐱}^{𝝂}+\text{polynomial of degree less than }n$. …In the co-monic basis ${\{Q_{𝝂}\}}_{|𝝂|=n}$, biorthogonal to the monic basis, $Q_{𝝂}$ is a polynomial of degree $n$ which is orthogonal to ${𝐱}^{𝝁}$ ($|𝝁|\le n$, $𝝁\ne 𝝂$) with respect to the inner product (37.13.1), analogous to (37.2.4). …

8: 1.11 Zeros of Polynomials

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§1.11(ii) Elementary Properties

… ►Every monic (coefficient of highest power is one) polynomial of odd degree with real coefficients has at least one real zero with sign opposite to that of the constant term. A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative. …

9: 37.14 Orthogonal Polynomials on the Simplex

… ►The monic basis ${\{V_{𝝂}^{𝜶}\}}_{|𝝂|=n}$ of ${𝒱}_n^{𝜶}({\mathrm{△}}^d)$ and the co-monic basis ${\{U_{𝝂}^{𝜶}\}}_{|𝝂|=n}$, biorthogonal to the monic basis, can be explicitly given by …Formula (37.14.9) is an analogue of the Rodrigues formulas in §18.5(ii). …

10: 37.3 Triangular Region with Weight Function

… ►The monic basis of ${𝒱}_n^{\alpha ,\beta ,\gamma }$ and the co-monic basis , biorthogonal to the monic basis, can be explicitly given as follows. … ►

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

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37.3.13

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Symbols:

${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , : co-monic basis element of two-variable Jacobi polynomial on the triangular region, : inner product on triangular region, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proveds:

Engelis (1964); Fackerell and Littler (1974, (11)); Dunkl and Xu (2014, Proposition 5.3.3); for $d=2$; Appell and Kampé de Fériet (1926, §XXXVI); only for (proved)

Source:

Erdélyi et al. (1953b, 12.4(17)); only for

Permalink:

http://dlmf.nist.gov/37.3.E13

Encodings:

pMML, png, TeX

See also:

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

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37.3.23

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $y$: real variable, : co-monic basis element of two-variable Jacobi polynomial on the triangular region, : 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

Encodings:

pMML, png, TeX

See also:

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

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37.3.25

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $(a,b)$: open interval, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $y$: real variable, : co-monic basis element of two-variable Jacobi polynomial on the triangular region, : 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

Encodings:

pMML, png, TeX

See also:

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

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