ultraspherical

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1: 18.3 Definitions

§18.3 Definitions

… ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of

x-1

for Jacobi polynomials, in powers of

x

for the other cases). …For explicit power series coefficients up to

n=12

for these polynomials and for coefficients up to

n=6

for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …

2: 18.6 Symmetry, Special Values, and Limits to Monomials

… ►For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. … ►

Table 18.6.1: Classical OP’s: symmetry and special values.

► ►►►

$p_{n}(x)$	$p_{n}(-x)$	$p_{n}(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime}(0)$
…
$C^{(\lambda)}_{n}\left(x\right)$	$(-1)^{n}C^{(\lambda)}_{n}\left(x\right)$	$\ifrac{{\left(2\lambda\right)_{n}}}{n!}$	$\ifrac{(-1)^{n}{\left(\lambda\right)_{n}}}{n!}$	$\ifrac{2(-1)^{n}{\left(\lambda\right)_{n+1}}}{n!}$
…

► ►

§18.6(ii) Limits to Monomials

… ►

18.6.4

\lim_{\lambda\to\infty}\frac{C^{(\lambda)}_{n}\left(x\right)}{C^{(\lambda)}_{n% }\left(1\right)}=x^{n},

ⓘ

Symbols:: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.18(iv), §18.6(ii)
Permalink:: http://dlmf.nist.gov/18.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.6(ii), §18.6 and Ch.18

…

3: 18.7 Interrelations and Limit Relations

… ►

Ultraspherical and Jacobi

… ►

Chebyshev, Ultraspherical, and Jacobi

… ►

Legendre, Ultraspherical, and Jacobi

… ►

Ultraspherical $\to$ Hermite

… ►

Ultraspherical $\to$ Chebyshev

…

4: 18.1 Notation

… ►

Ultraspherical (or Gegenbauer): $C^{(\lambda)}_{n}\left(x\right)$ .

… ►

Continuous $q$ -Ultraspherical: $C_{n}\left(x;\beta\,|\,q\right)$ .

… ►In Szegő (1975, §4.7) the ultraspherical polynomials

C^{(\lambda)}_{n}\left(x\right)

are denoted by

P_{n}^{(\lambda)}(x)

. The ultraspherical polynomials will not be considered for

\lambda=0

. They are defined in the literature by

C^{(0)}_{0}\left(x\right)=1

and …

5: 18.9 Recurrence Relations and Derivatives

… ►

Ultraspherical

►

18.9.7

(n+\lambda)C^{(\lambda)}_{n}\left(x\right)=\lambda\left(C^{(\lambda+1)}_{n}% \left(x\right)-C^{(\lambda+1)}_{n-2}\left(x\right)\right),

ⓘ

Symbols:: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

►

18.9.8

4\lambda(n+\lambda+1)(1-x^{2})C^{(\lambda+1)}_{n}\left(x\right)=-(n+1)(n+2)C^{% (\lambda)}_{n+2}\left(x\right)+(n+2\lambda)(n+2\lambda+1)C^{(\lambda)}_{n}% \left(x\right).

ⓘ

Symbols:: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.9.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

… ►

Ultraspherical

►

18.9.19

\frac{\mathrm{d}}{\mathrm{d}x}C^{(\lambda)}_{n}\left(x\right)=2\lambda C^{(% \lambda+1)}_{n-1}\left(x\right),

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §1.10(xi), §18.17(i), §18.9(iii)
Permalink:: http://dlmf.nist.gov/18.9.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

…

6: 18.10 Integral Representations

… ►

Ultraspherical

►

18.10.1

\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma% \left(\alpha+1\right)}{{\pi}^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right% )}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{% 2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\,\mathrm{d}\phi,

0<\theta<\pi

\alpha>-\tfrac{1}{2}

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial and $n$ : nonnegative integer
A&S Ref:: 22.10.11
Referenced by:: §18.10(i), §18.10(i)
Permalink:: http://dlmf.nist.gov/18.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

… ►

Ultraspherical

►

18.10.4

{\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}}=\frac{\Gamma\left(\alpha+1\right)}{{% \pi}^{\frac{1}{2}}\Gamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos\theta+i% \sin\theta\cos\phi)^{n}\*(\sin\phi)^{2\alpha}\,\mathrm{d}\phi},

\alpha>-\frac{1}{2}

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial and $n$ : nonnegative integer
A&S Ref:: 22.10.10
Referenced by:: §18.10(ii)
Permalink:: http://dlmf.nist.gov/18.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

… ►

Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

► ►►

$p_{n}(x)$	$g_{0}(x)$	$g_{1}(z,x)$	$g_{2}(z,x)$	$c$	Conditions
…

► …

7: 18.18 Sums

… ►

Ultraspherical

… ►

Ultraspherical

… ►

Ultraspherical

… ►

Ultraspherical

►

18.18.29

\sum_{\ell=0}^{n}C^{(\lambda)}_{\ell}\left(x\right)C^{(\mu)}_{n-\ell}\left(x% \right)=C^{(\lambda+\mu)}_{n}\left(x\right),

ⓘ

Symbols:: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.18(viii)
Permalink:: http://dlmf.nist.gov/18.18.E29
Encodings:: pMML, png, TeX
See also:: Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

…

8: 18.8 Differential Equations

… ►

Table 18.8.1: Classical OP’s: differential equations

A(x)f^{\prime\prime}(x)+B(x)f^{\prime}(x)+C(x)f(x)+\lambda_{n}f(x)=0

► ►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
…
4	$C^{(\lambda)}_{n}\left(x\right)$	$1-x^{2}$	$-(2\lambda+1)x$	$0$	$n(n+2\lambda)$
…

► …

9: 18.17 Integrals

… ►

Ultraspherical

… ►

Ultraspherical

… ►

Ultraspherical

… ►

Ultraspherical

… ►

Ultraspherical

…

10: 18.12 Generating Functions

… ►The

z

-radii of convergence will depend on

x

, and in first instance we will assume

x\in[-1,1]

for Jacobi, ultraspherical, Chebyshev and Legendre,

x\in[0,\infty)

for Laguerre, and

x\in\mathbb{R}

for Hermite. … ►

Ultraspherical

►

18.12.4

(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{(\lambda)}_{n}\left(x\right)z^{% n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\tfrac% {1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x% \right)z^{n},

|z|<1

ⓘ

Symbols:: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.9.3
Referenced by:: §18.12, §18.18(viii)
Permalink:: http://dlmf.nist.gov/18.12.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.12, §18.12 and Ch.18

►

18.12.5

\frac{1-xz}{(1-2xz+z^{2})^{\lambda+1}}=\sum_{n=0}^{\infty}\frac{n+2\lambda}{2% \lambda}C^{(\lambda)}_{n}\left(x\right)z^{n},

|z|<1

ⓘ

Symbols:: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.12, §18.18(viii)
Permalink:: http://dlmf.nist.gov/18.12.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.12, §18.12 and Ch.18

►

18.12.6

\Gamma\left(\lambda+\tfrac{1}{2}\right){\mathrm{e}}^{z\cos\theta}(\tfrac{1}{2}% z\sin\theta)^{\frac{1}{2}-\lambda}J_{\lambda-\frac{1}{2}}\left(z\sin\theta% \right)=\sum_{n=0}^{\infty}\frac{C^{(\lambda)}_{n}\left(\cos\theta\right)}{{% \left(2\lambda\right)_{n}}}z^{n},

0\leq\theta\leq\pi

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §18.12
Permalink:: http://dlmf.nist.gov/18.12.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §18.12, §18.12 and Ch.18

…

ultraspherical

1—10 of 33 matching pages

1: 18.3 Definitions

§18.3 Definitions

2: 18.6 Symmetry, Special Values, and Limits to Monomials

§18.6(ii) Limits to Monomials

3: 18.7 Interrelations and Limit Relations

Ultraspherical and Jacobi

Chebyshev, Ultraspherical, and Jacobi

Legendre, Ultraspherical, and Jacobi

Ultraspherical → Hermite

Ultraspherical → Chebyshev

4: 18.1 Notation

5: 18.9 Recurrence Relations and Derivatives

Ultraspherical

Ultraspherical

6: 18.10 Integral Representations

Ultraspherical

Ultraspherical

7: 18.18 Sums

Ultraspherical

Ultraspherical

Ultraspherical

Ultraspherical

8: 18.8 Differential Equations

9: 18.17 Integrals

Ultraspherical

Ultraspherical

Ultraspherical

Ultraspherical

Ultraspherical

10: 18.12 Generating Functions

Ultraspherical

Ultraspherical $\to$ Hermite

Ultraspherical $\to$ Chebyshev