complex orthogonal polynomials

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21: Errata

… ►The specific updates to Chapter 18 include some results for general orthogonal polynomials including quadratic transformations, uniqueness of orthogonality measure and completeness, moments, continued fractions, and some special classes of orthogonal polynomials. …We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. …We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials. We have also completely expanded our discussion on applications of orthogonal polynomials in the physical sciences, and also methods of computation for orthogonal polynomials. … ►

The reference Ismail (2005) has been replaced throughout by the further corrected paperback version Ismail (2009).

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22: 13.6 Relations to Other Functions

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§13.6(v) Orthogonal Polynomials

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Hermite Polynomials

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Laguerre Polynomials

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13.6.19 $$U(-n,\alpha +1,z)={(-1)}^n{\left(\alpha +1\right)}_{n}M(-n,\alpha +1,z)={(-1)}^{n}n!L_n^{(\alpha )}\left(z\right).$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.6.9 and 13.6.27

Referenced by:

§13.6(v), §18.17(iv), §18.9(iii)

Permalink:

http://dlmf.nist.gov/13.6.E19

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See also:

Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

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Charlier Polynomials

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23: 18.25 Wilson Class: Definitions

§18.25 Wilson Class: Definitions

… ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. … ►Under certain conditions on their parameters the orthogonality range for the Wilson polynomials and continuous dual Hahn polynomials is $(0,\mathrm{\infty })\cup S$, where $S$ is a specific finite set, e. … ►

Further Constraints for Racah Polynomials

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§18.25(ii) Weights and Standardizations: Continuous Cases

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24: Bibliography D

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Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, Vol. 3, New York University Courant Institute of Mathematical Sciences, New York.

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External Links:

ISBN 0-9658703-2-4; 0-8218-2695-6, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§18.38(ii), §18.38(ii).

Encodings:

BibTeX

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Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 (12), pp. 1491–1552.

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External Links:

ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

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Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory 162 (10), pp. 1793–1804.

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External Links:

ISSN 0021-9045, Document, FullText, MathReview (Francisco Marcellán), ZentralBlatt

Referenced by:

(18.16.12), (18.16.13), §18.16(ii), §18.16(ii), §18.16(iv).

Encodings:

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Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal. 30 (5), pp. 1029–1056.

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External Links:

ISSN 0036-1410, Document, MathReview (Kathi Karima Selig), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

BibTeX

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Inequalities for extreme zeros of some classical orthogonal and $q$-orthogonal polynomials. Math. Model. Nat. Phenom. 8 (1), pp. 48–59.

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External Links:

ISSN 0973-5348, Document, FullText, MathReview (Ana Foulquié Moreno), ZentralBlatt

Referenced by:

§18.16(ii), §18.16(iv), §18.16(ii), §18.16(iv), §18.27(iv), §18.27(iv), §18.27(v), §18.27(v).

Encodings:

BibTeX

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25: 18.23 Hahn Class: Generating Functions

§18.23 Hahn Class: Generating Functions

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Hahn

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18.23.3 $${\left(1-\frac{1-p}{p}z\right)}^x{(1+z)}^{N-x}=\sum _{n=0}^N\left(\genfrac{}{}{0.0pt}{}{N}{n}\right)K_n(x;p,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

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Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $N$: positive integer, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii), §18.23

Permalink:

http://dlmf.nist.gov/18.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

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18.23.5 $${\mathrm{e}}^z{\left(1-\frac{z}{a}\right)}^x=\sum _{n=0}^{\mathrm{\infty }}\frac{C_n(x;a)}{n!}{z}^n,$$ $x=0,1,2,\mathrm{\dots }$.

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Symbols:

$C_n(x;a)$: Charlier polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii), §18.23

Permalink:

http://dlmf.nist.gov/18.23.E5

Encodings:

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See also:

Annotations for §18.23, §18.23 and Ch.18

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18.23.7 $${(1-{\mathrm{e}}^{\mathrm{i}\varphi }z)}^{-\lambda +\mathrm{i}x}{(1-{\mathrm{e}}^{-\mathrm{i}\varphi }z)}^{-\lambda -\mathrm{i}x}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{(\lambda )}(x;\varphi )z^n,$$ .

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii), §18.23

Permalink:

http://dlmf.nist.gov/18.23.E7

Encodings:

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See also:

Annotations for §18.23, §18.23 and Ch.18

26: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to \mathrm{\infty }$ ) of the Hahn polynomials $Q_n(z;\alpha ,\beta ,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. … ►Corresponding approximations are included for the zeros of $P_n^{(\lambda )}(nx;\varphi )$. ►

Approximations in Terms of Laguerre Polynomials

… ►Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.

27: 13.18 Relations to Other Functions

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§13.18(v) Orthogonal Polynomials

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Hermite Polynomials

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13.18.14 $$M_{\frac{1}{4}+n,-\frac{1}{4}}\left(z^2\right)={(-1)}^n\frac{n!}{(2n)!}{\mathrm{e}}^{-\frac{1}{2}{z}^2}\sqrt{z}{H}_{2n}\left(z\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E14

Encodings:

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See also:

Annotations for §13.18(v), §13.18(v), §13.18 and Ch.13

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Laguerre Polynomials

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13.18.17 $$W_{\frac{1}{2}\alpha +\frac{1}{2}+n,\frac{1}{2}\alpha }\left(z\right)={(-1)}^n{\left(\alpha +1\right)}_n{M}_{\frac{1}{2}\alpha +\frac{1}{2}+n,\frac{1}{2}\alpha }\left(z\right)={(-1)}^{n}n!{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}\alpha +\frac{1}{2}}{L}_n^{(\alpha )}\left(z\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

Referenced by:

(33.14.14)

Permalink:

http://dlmf.nist.gov/13.18.E17

Encodings:

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See also:

Annotations for §13.18(v), §13.18(v), §13.18 and Ch.13

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28: Bibliography B

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Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 25–53.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.

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External Links:

ISSN 0003-486X, Document, MathReview (Thomas Kriecherbauer), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

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A uniform asymptotic formula for orthogonal polynomials associated with $\exp (-x^4)$ . J. Approx. Theory 98, pp. 146–166.

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External Links:

ISSN 0021-9045, Document, MathReview (V. L. Deshpande), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

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Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.

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External Links:

ISBN 3-7643-1100-2, MathReview (A. Magnus), ZentralBlatt

Referenced by:

§3.11(iv).

Encodings:

BibTeX

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Josef Meixner: his life and his orthogonal polynomials. Indag. Math. (N.S.) 30 (1), pp. 250–264.

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External Links:

ISSN 0019-3577, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

BibTeX

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29: 10.54 Integral Representations

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10.54.1 $${𝗃}_n\left(z\right)=\frac{z^n}{2^{n+1}n!}{\int }_0^{\pi }\cos \left(z\cos \theta \right){(\sin \theta )}^{2n+1}d\theta .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.13

Referenced by:

§10.54

Permalink:

http://dlmf.nist.gov/10.54.E1

Encodings:

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See also:

Annotations for §10.54 and Ch.10

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10.54.2 $${𝗃}_n\left(z\right)=\frac{{(-\mathrm{i})}^n}{2}{\int }_0^{\pi }{\mathrm{e}}^{\mathrm{i}z\cos \theta }{P}_n\left(\cos \theta \right)\sin \theta d\theta .$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.14

Permalink:

http://dlmf.nist.gov/10.54.E2

Encodings:

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See also:

Annotations for §10.54 and Ch.10

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10.54.3 $${𝗄}_n\left(z\right)=\frac{\pi }{2}{\int }_1^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{P}_n\left(t\right)dt,$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.54.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

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10.54.4 $${𝗃}_n\left(z\right)=\frac{{(-\mathrm{i})}^{n+1}}{2\pi }{\int }_{\mathrm{i}\mathrm{\infty }}^{(-1+,1+)}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b)$: open interval, $\mathrm{ph}$: phase, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.54.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

… ►For the Legendre polynomial $P_n$ and the associated Legendre function $Q_n$ see §§18.3 and 14.21(i), with $\mu =0$ and $\nu =n$. …

30: 15.9 Relations to Other Functions

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§15.9(i) Orthogonal Polynomials

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Jacobi

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Legendre

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Krawtchouk

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Meixner

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