polynomials orthogonal on the unit circle

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21: 10.60 Sums

… ►Then with $P_n$ again denoting the Legendre polynomial of degree $n$, ►

10.60.1 $$\frac{\cos w}{w}=-\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗃}_n\left(v\right){𝗒}_n\left(u\right)P_n\left(\cos \alpha \right),$$ .

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind and $n$: integer

A&S Ref:

10.1.46 (corrected)

Referenced by:

§10.60(i)

Permalink:

http://dlmf.nist.gov/10.60.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(i), §10.60 and Ch.10

►

10.60.2 $$\frac{\sin w}{w}=\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗃}_n\left(v\right){𝗃}_n\left(u\right)P_n\left(\cos \alpha \right).$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer

A&S Ref:

10.1.45

Referenced by:

§10.60(iii)

Permalink:

http://dlmf.nist.gov/10.60.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(i), §10.60 and Ch.10

►

10.60.3 $$\frac{{\mathrm{e}}^{-w}}{w}=\frac{2}{\pi }\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗂}_n^{(1)}\left(v\right){𝗄}_n\left(u\right)P_n\left(\cos \alpha \right),$$ .

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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, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${𝗄}_n\left(z\right)$: modified spherical Bessel function and $n$: integer

A&S Ref:

10.2.35 (with condition added)

Referenced by:

§10.60(i)

Permalink:

http://dlmf.nist.gov/10.60.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(i), §10.60 and Ch.10

… ►

10.60.7 $${\mathrm{e}}^{\mathrm{i}z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}(2n+1){\mathrm{i}}^n{𝗃}_n\left(z\right)P_n\left(\cos \alpha \right),$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.47

Referenced by:

§10.60(iii), Ch.10

Permalink:

http://dlmf.nist.gov/10.60.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(iii), §10.60 and Ch.10

…

22: 14.30 Spherical and Spheroidal Harmonics

… ► $P_n^m\left(\mathrm{i}x\right)$ and $Q_n^m\left(\mathrm{i}x\right)$ ($x>0$) are known as oblate spheroidal harmonics of the first and second kinds, respectively. Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics $R_n^m(x)={\mathrm{e}}^{-\mathrm{i}\pi n/2}{P}_n^m\left(\mathrm{i}x\right)$ and $T_n^m(x)=\mathrm{i}{\mathrm{e}}^{\mathrm{i}\pi n/2}{Q}_n^m\left(\mathrm{i}x\right)$ which are real when $x>0$ and $n=0,1,2,\mathrm{\dots }$. … ►

Orthogonality

… ►In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. See Andrews et al. (1999, Chapter 9). …

23: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► $u_{\lambda }\in 𝒟(T)$, corresponding to distinct eigenvalues, are orthogonal: i. … ►, of unit multiplicity, unless otherwise stated. … ►Orthogonality and normalization may then be chosen such that analogous to (1.18.19) and (1.18.20), we have … ►Then orthogonality and normalization relations are … ►Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal. …

24: 3.11 Approximation Techniques

… ►

§3.11(i) Minimax Polynomial Approximations

… ►They enjoy an orthogonal property with respect to integrals: …as well as an orthogonal property with respect to sums, as follows. When $n>0$ and $0\le j\le n$, $0\le k\le n$, … ► …

25: 28.31 Equations of Whittaker–Hill and Ince

… ►

§28.31(ii) Equation of Ince; Ince Polynomials

… ►When $p$ is a nonnegative integer, the parameter $\eta$ can be chosen so that solutions of (28.31.3) are trigonometric polynomials, called Ince polynomials. … … ►The normalization is given by … ►More important are the double orthogonality relations for $p_1\ne p_2$ or $m_1\ne m_2$ or both, given by …