complex orthogonal polynomials

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1: 18.37 Classical OP’s in Two or More Variables

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18.37.1 $$R_{m,n}^{(\alpha )}\left(r{\mathrm{e}}^{\mathrm{i}\theta }\right)={\mathrm{e}}^{\mathrm{i}(m-n)\theta }{r}^{|m-n|}\frac{P_{\min (m,n)}^{(\alpha ,|m-n|)}\left(2r^2-1\right)}{P_{\min (m,n)}^{(\alpha ,|m-n|)}\left(1\right)},$$ $r\ge 0$, $\theta \in ℝ$, $\alpha >-1$.

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Defines:

$R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $ℝ$: real line, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.37(ii)

Permalink:

http://dlmf.nist.gov/18.37.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.3 $$R_{m,n}^{(\alpha )}\left(z\right)=\sum _{j=0}^{\min (m,n)}{c}_j{z}^{m-j}{\overline{z}}^{n-j},$$

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Symbols:

$\overline{z}$: complex conjugate, $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.37.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.6 $$R_{m,n}^{(\alpha )}\left(z\right)=\sum _{j=0}^{\min (m,n)}\frac{{(-1)}^j{\left(\alpha +1\right)}_{m+n-j}{\left(-m\right)}_j{\left(-n\right)}_j}{{\left(\alpha +1\right)}_m{\left(\alpha +1\right)}_{n}j!}{z}^{m-j}{\overline{z}}^{n-j}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $!$: factorial (as in $n!$), $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.37.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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2: 3.5 Quadrature

… ►Complex orthogonal polynomials $p_n(1/\zeta )$ of degree $n=0,1,2,\mathrm{\dots }$, in $1/\zeta$ that satisfy the orthogonality condition … …

3: Software Index

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	Open Source													With Book						Commercial
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16.27(iii) Complex Arguments	✓											a	✓								✓	✓	✓
18 Orthogonal Polynomials
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22.22(iii) Complex Argument	✓					✓						✓	✓	✓						✓	✓	✓	a	✓
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23.24(iii) Complex Argument														✓						✓	✓	✓	a
24 Bernoulli and Euler Polynomials
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4: 18.33 Polynomials Orthogonal on the Unit Circle

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18.33.8 $$p_n\left(\frac{1}{2}(z+z^{-1})\right)=\text{(const.)}\times \left(z^{-n}{\varphi }_{2n}(z)+z^n{\varphi }_{2n}(z^{-1})\right)=\text{(const.)}\times \left(z^{-n+1}{\varphi }_{2n-1}(z)+z^{n-1}{\varphi }_{2n-1}(z^{-1})\right),$$

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Symbols:

$z$: complex variable, $n$: nonnegative integer, $p_n(x)$: an orthogonal polynomial and ${\varphi }_n(z)$: polynomials

Referenced by:

§18.33(iii)

Permalink:

http://dlmf.nist.gov/18.33.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(iii), §18.33 and Ch.18

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18.33.9 $$q_n\left(\frac{1}{2}(z+z^{-1})\right)=\text{(const.)}\times \frac{z^{-n-1}{\varphi }_{2n+2}(z)-z^{n+1}{\varphi }_{2n+2}(z^{-1})}{z-z^{-1}}=\text{(const.)}\times \frac{z^{-n}{\varphi }_{2n+1}(z)-z^n{\varphi }_{2n+1}(z^{-1})}{z-z^{-1}}.$$

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Symbols:

$z$: complex variable, $n$: nonnegative integer, $q_n(x)$: an orthogonal polynomial and ${\varphi }_n(z)$: polynomials

Referenced by:

§18.33(iii), §18.33(iii)

Permalink:

http://dlmf.nist.gov/18.33.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(iii), §18.33 and Ch.18

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18.33.10 $$z^{-n}{\varphi }_{2n}(z)=A_n{p}_n\left(\frac{1}{2}(z+z^{-1})\right)+B_n(z-z^{-1})q_{n-1}\left(\frac{1}{2}(z+z^{-1})\right),$$

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Symbols:

$z$: complex variable, $n$: nonnegative integer, $q_n(x)$: an orthogonal polynomial, $p_n(x)$: an orthogonal polynomial, $A_n$: coefficient, $B_n$: coefficient and ${\varphi }_n(z)$: polynomials

Referenced by:

§18.33(iii)

Permalink:

http://dlmf.nist.gov/18.33.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(iii), §18.33 and Ch.18

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18.33.11 $$z^{-n+1}{\varphi }_{2n-1}(z)=C_n{p}_n\left(\frac{1}{2}(z+z^{-1})\right)+D_n(z-z^{-1})q_{n-1}\left(\frac{1}{2}(z+z^{-1})\right),$$

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Symbols:

$z$: complex variable, $n$: nonnegative integer, $q_n(x)$: an orthogonal polynomial, $p_n(x)$: an orthogonal polynomial, $C_n$: coefficient, $D_n$: coefficient and ${\varphi }_n(z)$: polynomials

Referenced by:

§18.33(iii)

Permalink:

http://dlmf.nist.gov/18.33.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(iii), §18.33 and Ch.18

… ►This states that for any sequence ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}$ with ${\alpha }_n\in ℂ$ and the polynomials ${\mathrm{\Phi }}_n(z)$ generated by the recurrence relations (18.33.23), (18.33.24) with ${\mathrm{\Phi }}_0(z)=1$ satisfy the orthogonality relation (18.33.17) for a unique probability measure $\mu$ with infinite support on the unit circle. …

5: Bibliography K

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Orthogonal polynomials and computer algebra. In Recent developments in complex analysis and computer algebra (Newark, DE, 1997), R. P. Gilbert, J. Kajiwara, and Y. S. Xu (Eds.), Int. Soc. Anal. Appl. Comput., Vol. 4, Dordrecht, pp. 205–234.

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External Links:

FullText, MathReview Entry, ZentralBlatt

Referenced by:

CAOP (website).

Encodings:

BibTeX

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6: 18.19 Hahn Class: Definitions

… ►These polynomials are orthogonal on $(-\mathrm{\infty },\mathrm{\infty })$, and with $\mathrm{\Re }a>0$, $\mathrm{\Re }b>0$ are defined as follows. …

7: 31.15 Stieltjes Polynomials

§31.15 Stieltjes Polynomials

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§31.15(i) Definitions

►Stieltjes polynomials are polynomial solutions of the Fuchsian equation (31.14.1). … … ►are mutually orthogonal over the set $Q$: …

8: 18.2 General Orthogonal Polynomials

… ►It is assumed throughout this chapter that for each polynomial $p_n(x)$ that is orthogonal on an open interval $(a,b)$ the variable $x$ is confined to the closure of $(a,b)$ unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) …

9: 18.38 Mathematical Applications

… ►also the case $\beta =0$ of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. …

10: 2.8 Differential Equations with a Parameter

… ►in which $u$ is a real or complex parameter, and asymptotic solutions are needed for large $|u|$ that are uniform with respect to $z$ in a point set $𝐃$ in $ℝ$ or $ℂ$. For example, $u$ can be the order of a Bessel function or degree of an orthogonal polynomial. … ►For error bounds, extensions to pure imaginary or complex $u$, an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10). … ►For error bounds, more delicate error estimates, extensions to complex $\xi$ and $u$, zeros, connection formulas, extensions to inhomogeneous equations, and examples, see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid (1974a, b), Boyd (1987), and Baldwin (1991). … ►For error bounds, more delicate error estimates, extensions to complex $\xi$, $\nu$, and $u$, zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a). …