ultraspherical polynomials

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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. ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
…
Ultraspherical (Gegenbauer)	$C_n^{(\lambda )}\left(x\right)$	$(-1,1)$	${(1-x^2)}^{\lambda -\frac{1}{2}}$	${\displaystyle \frac{2^{1-2\lambda }\pi \mathrm{\Gamma }\left(n+2\lambda \right)}{(n+\lambda ){\left(\mathrm{\Gamma }\left(\lambda \right)\right)}^{2}n!}}$	${\displaystyle \frac{2^n{\left(\lambda \right)}_n}{n!}}$	$0$	$\lambda >-\frac{1}{2},\lambda \ne 0$
…

►

►For exact values of the coefficients of the Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$, the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$, the Chebyshev polynomials $T_n\left(x\right)$ and $U_n\left(x\right)$, the Legendre polynomials $P_n\left(x\right)$, the Laguerre polynomials $L_n\left(x\right)$, and the Hermite polynomials $H_n\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). …The ultraspherical polynomials are in powers of $x$ for $n=0,1,\mathrm{\dots },6$. …

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.

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

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►

§18.6(ii) Limits to Monomials

… ►

18.6.4 $$\underset{\lambda \to \mathrm{\infty }}{lim}\frac{C_n^{(\lambda )}\left(x\right)}{C_n^{(\lambda )}\left(1\right)}=x^n,$$

ⓘ

Symbols:

$C_n^{(\lambda )}\left(x\right)$: 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

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3: 18.9 Recurrence Relations and Derivatives

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18.9.7 $$(n+\lambda )C_n^{(\lambda )}\left(x\right)=\lambda \left(C_n^{(\lambda +1)}\left(x\right)-C_{n-2}^{(\lambda +1)}\left(x\right)\right),$$

ⓘ

Symbols:

$C_n^{(\lambda )}\left(x\right)$: 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 $$4\lambda (n+\lambda +1)(1-x^2)C_n^{(\lambda +1)}\left(x\right)=-(n+1)(n+2)C_{n+2}^{(\lambda )}\left(x\right)+(n+2\lambda )(n+2\lambda +1)C_n^{(\lambda )}\left(x\right).$$

ⓘ

Symbols:

$C_n^{(\lambda )}\left(x\right)$: 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 $$\frac{d}{dx}{C}_n^{(\lambda )}\left(x\right)=2\lambda C_{n-1}^{(\lambda +1)}\left(x\right),$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§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

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18.9.20 $$\frac{d}{dx}\left({(1-x^2)}^{\lambda -\frac{1}{2}}{C}_n^{(\lambda )}\left(x\right)\right)=-\frac{(n+1)(n+2\lambda -1)}{2(\lambda -1)}{(1-x^2)}^{\lambda -\frac{3}{2}}{C}_{n+1}^{(\lambda -1)}\left(x\right).$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.9(iii)

Permalink:

http://dlmf.nist.gov/18.9.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

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4: 18.7 Interrelations and Limit Relations

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18.7.1 $$C_n^{(\lambda )}\left(x\right)=\frac{{\left(2\lambda \right)}_n}{{\left(\lambda +\frac{1}{2}\right)}_n}{P}_n^{(\lambda -\frac{1}{2},\lambda -\frac{1}{2})}\left(x\right),$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.27

Referenced by:

§18.15(ii), §18.7(i), §18.7(ii), §18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

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18.7.9 $$P_n\left(x\right)=C_n^{(\frac{1}{2})}\left(x\right)=P_n^{(0,0)}\left(x\right).$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $P_n\left(x\right)$: Legendre polynomial, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.35, 22.5.36

Referenced by:

§10.60(iii), §18.18(ii), §18.5(iii)

Permalink:

http://dlmf.nist.gov/18.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

… ►

18.7.15 $$C_{2n}^{(\lambda )}\left(x\right)=\frac{{\left(\lambda \right)}_n}{{\left(\frac{1}{2}\right)}_n}{P}_n^{(\lambda -\frac{1}{2},-\frac{1}{2})}\left(2x^2-1\right),$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.25

Referenced by:

§18.7(ii), §18.9(ii)

Permalink:

http://dlmf.nist.gov/18.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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18.7.24 $$\underset{\lambda \to \mathrm{\infty }}{lim}{\lambda }^{-\frac{1}{2}n}{C}_n^{(\lambda )}\left({\lambda }^{-\frac{1}{2}}x\right)=\frac{H_n\left(x\right)}{n!}.$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.15.6

Referenced by:

§18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E24

Encodings:

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See also:

Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

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18.7.25 $$\underset{\lambda \to 0}{lim}\frac{1}{\lambda }{C}_n^{(\lambda )}\left(x\right)=\frac{2}{n}{T}_n\left(x\right),$$ $n\ge 1$.

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.7(iii)

Permalink:

http://dlmf.nist.gov/18.7.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

…

5: 18.18 Sums

… ►

Ultraspherical

… ►

Ultraspherical

… ►

Ultraspherical

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18.18.29 $$\sum _{\mathrm{\ell }=0}^n{C}_{\mathrm{\ell }}^{(\lambda )}\left(x\right)C_{n-\mathrm{\ell }}^{(\mu )}\left(x\right)=C_n^{(\lambda +\mu )}\left(x\right).$$

ⓘ

Symbols:

$C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.18(viii)

Permalink:

http://dlmf.nist.gov/18.18.E29

Encodings:

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See also:

Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

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18.18.30 $$\sum _{\mathrm{\ell }=0}^n\frac{\mathrm{\ell }+2\lambda }{2\lambda }{C}_{\mathrm{\ell }}^{(\lambda )}\left(x\right)x^{n-\mathrm{\ell }}=C_n^{(\lambda +1)}\left(x\right).$$

ⓘ

Symbols:

$C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.18(viii)

Permalink:

http://dlmf.nist.gov/18.18.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(viii), §18.18(viii), §18.18 and Ch.18

…

6: 18.1 Notation

… ►

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

… ►

Continuous $q$-Ultraspherical: $C_n(x;\beta |q)$.

… ►In Szegő (1975, §4.7) the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$ 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 …

7: 18.10 Integral Representations

… ►

Ultraspherical

►

18.10.1 $$\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{2^{\alpha +\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +\frac{1}{2}\right)}{(\sin \theta )}^{-2\alpha }{\int }_0^{\theta }\frac{\cos \left((n+\alpha +\frac{1}{2})\varphi \right)}{{(\cos \varphi -\cos \theta )}^{-\alpha +\frac{1}{2}}}d\varphi ,$$ , $\alpha >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential, $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: 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 $$\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\int }_0^{\pi }{(\cos \theta +\mathrm{i}\sin \theta \cos \varphi )}^n{(\sin \varphi )}^{2\alpha }d\varphi ,$$ $\alpha >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: 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).

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$p_n(x)$	$g_0(x)$	$g_1(z,x)$	$g_2(z,x)$	$c$	Conditions
…
$C_n^{(\lambda )}\left(x\right)$	$1$	$z^{-1}$	${(1-2xz+z^2)}^{-\lambda }$	$0$	${\mathrm{e}}^{\pm \mathrm{i}\theta }$ outside $C$ (where $x=\cos \theta$).
…

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…

8: 18.12 Generating Functions

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Ultraspherical

►

18.12.4 $${(1-2xz+z^2)}^{-\lambda }=\sum _{n=0}^{\mathrm{\infty }}{C}_n^{(\lambda )}\left(x\right)z^n=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(2\lambda \right)}_n}{{\left(\lambda +\frac{1}{2}\right)}_n}{P}_n^{(\lambda -\frac{1}{2},\lambda -\frac{1}{2})}\left(x\right)z^n,$$ .

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: 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

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18.12.5 $$\frac{1-xz}{{(1-2xz+z^2)}^{\lambda +1}}=\sum _{n=0}^{\mathrm{\infty }}\frac{n+2\lambda }{2\lambda }{C}_n^{(\lambda )}\left(x\right)z^n,$$ .

ⓘ

Symbols:

$C_n^{(\lambda )}\left(x\right)$: 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

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18.12.6 $$\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right){\mathrm{e}}^{z\cos \theta }{(\frac{1}{2}z\sin \theta )}^{\frac{1}{2}-\lambda }{J}_{\lambda -\frac{1}{2}}\left(z\sin \theta \right)=\sum _{n=0}^{\mathrm{\infty }}\frac{C_n^{(\lambda )}\left(\cos \theta \right)}{{\left(2\lambda \right)}_n}{z}^n,$$ $0\le \theta \le \pi$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: 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

…

9: 18.14 Inequalities

… ►

Ultraspherical

►

18.14.4 $$|C_n^{(\lambda )}\left(x\right)|\le C_n^{(\lambda )}\left(1\right)=\frac{{\left(2\lambda \right)}_n}{n!},$$ $-1\le x\le 1$, $\lambda >0$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.14.2

Referenced by:

§18.14(i)

Permalink:

http://dlmf.nist.gov/18.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

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18.14.5 $$|C_{2m}^{(\lambda )}\left(x\right)|\le |C_{2m}^{(\lambda )}\left(0\right)|=\left|\frac{{\left(\lambda \right)}_m}{m!}\right|,$$ $-1\le x\le 1$, ,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $m$: nonnegative integer and $x$: real variable

A&S Ref:

22.14.2

Referenced by:

§18.14(i)

Permalink:

http://dlmf.nist.gov/18.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

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18.14.6 $-1\le x\le 1$, .

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $m$: nonnegative integer and $x$: real variable

Referenced by:

§18.14(i)

Permalink:

http://dlmf.nist.gov/18.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

►

18.14.7 $-1\le x\le 1$, .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.14(i)

Permalink:

http://dlmf.nist.gov/18.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

…

10: 18.5 Explicit Representations

… ►

Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).

► ►►►

$p_n(x)$	$F(x)$	${\kappa }_n$
…
$C_n^{(\lambda )}\left(x\right)$	$1-x^2$	${\displaystyle \frac{{(-2)}^n{\left(\lambda +\frac{1}{2}\right)}_{n}n!}{{\left(2\lambda \right)}_n}}$
…

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… ►

18.5.9 $$C_n^{(\lambda )}\left(x\right)=\frac{{\left(2\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,n+2\lambda }{\lambda +\frac{1}{2}};\frac{1-x}{2}),$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.46

Referenced by:

§18.5(iii), §18.6(i)

Permalink:

http://dlmf.nist.gov/18.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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18.5.10 $$C_n^{(\lambda )}\left(x\right)=\sum _{\mathrm{\ell }=0}^{\lfloor n/2\rfloor }\frac{{(-1)}^{\mathrm{\ell }}{\left(\lambda \right)}_{n-\mathrm{\ell }}}{\mathrm{\ell }!(n-2\mathrm{\ell })!}{(2x)}^{n-2\mathrm{\ell }}={(2x)}^n\frac{{\left(\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}n,-\frac{1}{2}n+\frac{1}{2}}{1-\lambda -n};\frac{1}{x^2}),$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$ , $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.3.4

Referenced by:

§18.17(vii), §18.5(iii), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

►

18.5.11 $$C_n^{(\lambda )}\left(\cos \theta \right)=\sum _{\mathrm{\ell }=0}^n\frac{{\left(\lambda \right)}_{\mathrm{\ell }}{\left(\lambda \right)}_{n-\mathrm{\ell }}}{\mathrm{\ell }!(n-\mathrm{\ell })!}\cos \left((n-2\mathrm{\ell })\theta \right)={\mathrm{e}}^{\mathrm{i}n\theta }\frac{{\left(\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,\lambda }{1-\lambda -n};{\mathrm{e}}^{-2\mathrm{i}\theta }).$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

22.3.12

Referenced by:

§18.5(iii)

Permalink:

http://dlmf.nist.gov/18.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►Similarly in the cases of the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$ and the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ we assume that $\lambda >-\frac{1}{2},\lambda \ne 0$, and $\alpha >-1$, unless stated otherwise. …