Gegenbauer polynomials

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1: 18.6 Symmetry, Special Values, and Limits to Monomials

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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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… ►

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

ⓘ

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

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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

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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:

pMML, png, TeX

See also:

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

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18.7.25

ⓘ

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

Source:

Andrews et al. (1999, (6.4.13))

Referenced by:

§18.7(i), §18.7(iii), Erratum (V1.2.0) for Equation (18.7.25)

Permalink:

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

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

We included the case $n=0$.

See also:

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

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3: 18.9 Recurrence Relations and Derivatives

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Table 18.9.1: Classical OP’s: recurrence relations (18.9.1).

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$p_n(x)$	$A_n$	$B_n$	$C_n$
$C_n^{(\lambda )}\left(x\right)$	$\frac{2(n+\lambda )}{n+1}$	$0$	$\frac{n+2\lambda -1}{n+1}$
…

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… ►

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

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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

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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

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18.9.19 $$\frac{d}{dx}{C}_n^{(\lambda )}\left(x\right)=2\lambda C_{n-1}^{(\lambda +1)}\left(x\right),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $C_n^{(\lambda )}\left(x\right)$: 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

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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 of $f$ with respect to $x$ , $C_n^{(\lambda )}\left(x\right)$: 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: 18.1 Notation

… ►

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

… ►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 ►

18.1.1 $$C_n^{(0)}\left(x\right)=\frac{2}{n}{T}_n\left(x\right)=\frac{2(n-1)!}{{\left(\frac{1}{2}\right)}_n}{P}_n^{(-\frac{1}{2},-\frac{1}{2})}\left(x\right),$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: 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

Permalink:

http://dlmf.nist.gov/18.1.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.1(iii), §18.1 and Ch.18

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5: 18.10 Integral Representations

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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 of $x$ , $\int$: 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

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18.10.4 $$\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\int }_0^{\pi }{(\cos \theta +i\sin \theta \cos \varphi )}^n{(\sin \varphi )}^{2\alpha }d\varphi ,$$ $\alpha >-\frac{1}{2}$.

ⓘ

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 of $x$ , $\int$: 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

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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 $x=\cos \theta$).
…

►

…

6: 37.15 Orthogonal Polynomials on the Ball

… ►Recursively define ultraspherical polynomials on the ball $C_{𝝂}^{(\alpha +\frac{1}{2})}\left(𝐱\right)$ ($𝝂\in {\mathbb{N}}_0^d$, $𝐱\in {ℝ}^d$) by ►

37.15.4 $$C_{{\nu }_1,\mathrm{\dots },{\nu }_d}^{(\alpha +\frac{1}{2})}\left(x_1,\mathrm{\dots },x_d\right)=C_{{\nu }_1}^{(\alpha +{\nu }_2+\mathrm{\cdots }+{\nu }_d+\frac{1}{2}d)}(x_1){\left(1-x_1^2\right)}^{\frac{1}{2}({\nu }_2+\mathrm{\cdots }+{\nu }_d)}{C}_{{\nu }_2,\mathrm{\dots },{\nu }_d}^{(\alpha +\frac{1}{2})}\left(\frac{x_2}{\sqrt{1-x_1^2}},\mathrm{\dots },\frac{x_d}{\sqrt{1-x_1^2}}\right),$$

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Defines:

$C_{𝝂}^{(\alpha +\frac{1}{2})}\left(𝐱\right)$: ultraspherical polynomials on the ball

Symbols:

$C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝝂$: multi-index of $d$-dimensions, $\alpha$: real variable and $x$: real variable

Referenced by:

§37.15(ii)

Permalink:

http://dlmf.nist.gov/37.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(ii), §37.15 and Ch.37

►where the case $d=1$ is given by the ultraspherical polynomials $C_{{\nu }_1}^{(\alpha +\frac{1}{2})}\left(x_1\right)$ in one variable as defined in Table 18.3.1. …For $d=2$ the polynomial $C_{{\nu }_1,{\nu }_2}^{(\alpha +\frac{1}{2})}\left(x_1,x_2\right)$ as defined by (37.15.4) becomes the polynomial $C_{{\nu }_2,{\nu }_1+{\nu }_2}^{(\alpha +\frac{1}{2})}(x_1,x_2)$ as given by (37.4.5). … ►See Dunkl and Xu (2014, Proposition 5.2.2) for the explicit value of ${⟨C_{𝝂}^{(\alpha +\frac{1}{2})},C_{𝝂}^{(\alpha +\frac{1}{2})}⟩}_{\alpha }$. …

7: 37.4 Disk with Weight Function ${\left(1-x^2-y^2\right)}^{\alpha }$

… ►

37.4.5 $$C_{k,n}^{(\alpha +\frac{1}{2})}(x,y)=C_{n-k}^{(\alpha +k+1)}\left(x\right){(1-x^2)}^{\frac{1}{2}k}{C}_k^{(\alpha +\frac{1}{2})}\left(\frac{y}{\sqrt{1-x^2}}\right).$$

ⓘ

Defines:

$C_{k,n}^{(\alpha +\frac{1}{2})}(x,y)$: two-variable ultraspherical polynomial on the disk

Symbols:

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

Source:

Dunkl and Xu (2014, (2.3.1)); The mentioned value of the constant in (37.4.4) follows by (18.7.1).

Referenced by:

§37.15(ii), (37.4.29), §37.4(ii), §37.4(iii), §37.4(vi)

Permalink:

http://dlmf.nist.gov/37.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.4(ii), §37.4 and Ch.37

… ►

37.4.6 $${⟨C_{k,n}^{(\alpha +\frac{1}{2})},C_{j,m}^{(\alpha +\frac{1}{2})}⟩}_{\alpha }=h_{k,n}^{(\alpha +\frac{1}{2})}{\delta }_{n,m}{\delta }_{k,j},$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $C_{k,n}^{(\alpha +\frac{1}{2})}(x,y)$: two-variable ultraspherical polynomial on the disk, $n$: nonnegative integer, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $\alpha$: real parameter, ${⟨f,g⟩}_{\alpha }$: inner product on disk and $h_{k,n}^{(\alpha +\frac{1}{2})}$: orthogonality norm for two-variable ultraspherical polynomial on $𝔻$

Permalink:

http://dlmf.nist.gov/37.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §37.4(ii), §37.4 and Ch.37

… ►There is also an orthogonal basis of ${𝒱}_n^{\alpha }$ consisting of polynomials $C_{k,n}^{(\alpha +\frac{1}{2})}(y,x)$ ($k=0,1,\mathrm{\dots },n$). … ►For $\alpha =0$, by (18.7.4), $C_n^{(1)}=U_n$, the Chebyshev polynomial of the second kind. … ►

37.4.29 $$[(1-x^2-y^2)D_{yy}-2(\alpha +1)y{D}_y]C_{k,n}^{(\alpha +\frac{1}{2})}(x,y)=-k(k+2\alpha +1)C_{k,n}^{(\alpha +\frac{1}{2})}(x,y),$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $C_{k,n}^{(\alpha +\frac{1}{2})}(x,y)$: two-variable ultraspherical polynomial on the disk, $n$: nonnegative integer, $z$: complex number, $k$: nonnegative integer, $x$: real variable, $y$: real variable and $\alpha$: real parameter

Proof sketch:

From (37.4.5) and Table 18.8.1 we get $\left[(1-z^2)D_{zz}-2(\alpha +1)D_z+k(k+2\alpha +1)\right]C_{k,n}^{(\alpha +\frac{1}{2})}(x,z\sqrt{1-x^2})=0$.

Permalink:

http://dlmf.nist.gov/37.4.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §37.4(vi), §37.4 and Ch.37

…

8: 18.18 Sums

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18.18.16 $$C_n^{(\mu )}\left(x\right)=\sum _{\mathrm{\ell }=0}^{\lfloor n/2\rfloor }\frac{\lambda +n-2\mathrm{\ell }}{\lambda }\frac{{\left(\mu \right)}_{n-\mathrm{\ell }}}{{\left(\lambda +1\right)}_{n-\mathrm{\ell }}}\frac{{\left(\mu -\lambda \right)}_{\mathrm{\ell }}}{\mathrm{\ell }!}{C}_{n-2\mathrm{\ell }}^{(\lambda )}\left(x\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: 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

Referenced by:

§18.18(iv), §18.18(iv)

Permalink:

http://dlmf.nist.gov/18.18.E16

Encodings:

pMML, png, TeX

See also:

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

►

18.18.17 $${(2x)}^n=n!\sum _{\mathrm{\ell }=0}^{\lfloor n/2\rfloor }\frac{\lambda +n-2\mathrm{\ell }}{\lambda }\frac{1}{{\left(\lambda +1\right)}_{n-\mathrm{\ell }}\mathrm{\ell }!}{C}_{n-2\mathrm{\ell }}^{(\lambda )}\left(x\right).$$

ⓘ

Symbols:

${\left(a\right)}_n$: 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

Referenced by:

§18.18(iv)

Permalink:

http://dlmf.nist.gov/18.18.E17

Encodings:

pMML, png, TeX

See also:

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

… ►

18.18.22 $$C_m^{(\lambda )}\left(x\right)C_n^{(\lambda )}\left(x\right)=\sum _{\mathrm{\ell }=0}^{\min (m,n)}\frac{(m+n+\lambda -2\mathrm{\ell })(m+n-2\mathrm{\ell })!}{(m+n+\lambda -\mathrm{\ell })\mathrm{\ell }!(m-\mathrm{\ell })!(n-\mathrm{\ell })!}\frac{{\left(\lambda \right)}_{\mathrm{\ell }}{\left(\lambda \right)}_{m-\mathrm{\ell }}{\left(\lambda \right)}_{n-\mathrm{\ell }}{\left(2\lambda \right)}_{m+n-\mathrm{\ell }}}{{\left(\lambda \right)}_{m+n-\mathrm{\ell }}{\left(2\lambda \right)}_{m+n-2\mathrm{\ell }}}{C}_{m+n-2\mathrm{\ell }}^{(\lambda )}\left(x\right).$$

ⓘ

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, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.17.41_5), §18.17(viii), §18.18(v)

Permalink:

http://dlmf.nist.gov/18.18.E22

Encodings:

pMML, png, TeX

See also:

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

… ►

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:

pMML, png, TeX

See also:

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

►

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

…

9: 18.17 Integrals

… ►

18.17.5 $$\frac{C_n^{(\lambda )}\left(\cos {\theta }_1\right)}{C_n^{(\lambda )}\left(1\right)}\frac{C_n^{(\lambda )}\left(\cos {\theta }_2\right)}{C_n^{(\lambda )}\left(1\right)}=\frac{\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\lambda \right)}{\int }_0^{\pi }\frac{C_n^{(\lambda )}\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \varphi \right)}{C_n^{(\lambda )}\left(1\right)}{(\sin \varphi )}^{2\lambda -1}d\varphi ,$$ $\lambda >0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial and $n$: nonnegative integer

Referenced by:

§18.17(ii), §18.17(ii), §18.18(ii)

Permalink:

http://dlmf.nist.gov/18.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(ii), §18.17(ii), §18.17 and Ch.18

… ►

18.17.12 $$\frac{\mathrm{\Gamma }\left(\lambda -\mu \right)C_n^{(\lambda -\mu )}\left(x^{-\frac{1}{2}}\right)}{x^{\lambda -\mu +\frac{1}{2}n}}={\int }_x^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\lambda \right)C_n^{(\lambda )}\left(y^{-\frac{1}{2}}\right)}{y^{\lambda +\frac{1}{2}n}}\frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\lambda >\mu >0$, $x>0$,

ⓘ

Symbols:

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

Referenced by:

§18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

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18.17.13 $$\frac{x^{\frac{1}{2}n}{(x-1)}^{\lambda +\mu -\frac{1}{2}}}{\mathrm{\Gamma }\left(\lambda +\mu +\frac{1}{2}\right)}\frac{C_n^{(\lambda +\mu )}\left(x^{-\frac{1}{2}}\right)}{C_n^{(\lambda +\mu )}\left(1\right)}={\int }_1^x\frac{y^{\frac{1}{2}n}{(y-1)}^{\lambda -\frac{1}{2}}}{\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}\frac{C_n^{(\lambda )}\left(y^{-\frac{1}{2}}\right)}{C_n^{(\lambda )}\left(1\right)}\frac{{(x-y)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\mu >0$, $x>1$.

ⓘ

Symbols:

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

Referenced by:

§1.15(vi), §18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

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18.17.16_5 $${\int }_{-1}^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_n^{(\lambda )}\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx=\frac{2\pi {\mathrm{i}}^n\mathrm{\Gamma }\left(n+2\lambda \right)J_{n+\lambda }\left(y\right)}{n!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }},$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v), Erratum (V1.2.0) §18.17

Permalink:

http://dlmf.nist.gov/18.17.E16_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

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18.17.41_5 $${\int }_{-1}^1{C}_{\mathrm{\ell }}^{(\lambda )}\left(x\right)C_m^{(\lambda )}\left(x\right)C_n^{(\lambda )}\left(x\right){(1-x^2)}^{\lambda -\frac{1}{2}}dx=\frac{{\left(\lambda \right)}_{\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m-\frac{1}{2}n}{\left(\lambda \right)}_{\frac{1}{2}m+\frac{1}{2}n-\frac{1}{2}\mathrm{\ell }}{\left(\lambda \right)}_{\frac{1}{2}n+\frac{1}{2}\mathrm{\ell }-\frac{1}{2}m}{\left(2\lambda \right)}_{\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m+\frac{1}{2}n}\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)\sqrt{\pi }}{(\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m-\frac{1}{2}n)!(\frac{1}{2}m+\frac{1}{2}n-\frac{1}{2}\mathrm{\ell })!(\frac{1}{2}n+\frac{1}{2}\mathrm{\ell }-\frac{1}{2}m)!\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m+\frac{1}{2}n+1\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Andrews et al. (1999, Corollary 6.8.4); Straightforward from (18.18.22).(proved)

Referenced by:

§18.17(vii), Erratum (V1.2.0) §18.17

Permalink:

http://dlmf.nist.gov/18.17.E41_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

…

10: 18.12 Generating Functions

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

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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

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