Gegenbauer polynomials

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11: 18.5 Explicit Representations

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

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

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

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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

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

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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:

§18.18(iv), §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. …

12: 1.10 Functions of a Complex Variable

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1.10.28 $$F(x,\lambda ;z)={\left(1-2xz+z^2\right)}^{-\lambda }=\sum _{n=0}^{\mathrm{\infty }}{C}_n^{(\lambda )}\left(x\right)z^n,$$ .

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Symbols:

$C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $z$: variable, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.10.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(xi), §1.10(xi), §1.10 and Ch.1

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1.10.29 $$\sum _{n=0}^{\mathrm{\infty }}\frac{d}{dx}{C}_n^{(\lambda )}\left(x\right)z^n=2\lambda z{\left(1-2xz+z^2\right)}^{-\lambda -1}=\sum _{n=0}^{\mathrm{\infty }}2\lambda C_n^{(\lambda +1)}\left(x\right)z^{n+1},$$

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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, $z$: variable and $n$: nonnegative integer

Referenced by:

Erratum (V1.2.0) Section 1.10

Permalink:

http://dlmf.nist.gov/1.10.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(xi), §1.10(xi), §1.10 and Ch.1

►and hence $\frac{d}{dx}{C}_n^{(\lambda )}\left(x\right)=2\lambda C_{n-1}^{(\lambda +1)}\left(x\right)$, that is (18.9.19). The recurrence relation for $C_n^{(\lambda )}\left(x\right)$ in §18.9(i) follows from $(1-2xz+z^2)\frac{\partial }{\partial z}F(x,\lambda ;z)=2\lambda (x-z)F(x,\lambda ;z)$, and the contour integral representation for $C_n^{(\lambda )}\left(x\right)$ in §18.10(iii) is just (1.10.27).

13: 10.23 Sums

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10.23.8 $$\frac{{𝒞}_{\nu }\left(w\right)}{w^{\nu }}=2^{\nu }\mathrm{\Gamma }\left(\nu \right)\sum _{k=0}^{\mathrm{\infty }}(\nu +k)\frac{{𝒞}_{\nu +k}\left(u\right)}{u^{\nu }}\frac{J_{\nu +k}\left(v\right)}{v^{\nu }}{C}_k^{(\nu )}\left(\cos \alpha \right),$$ $\nu \ne 0,-1,\mathrm{\dots }$, ,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\cos z$: cosine function, ${𝒞}_{\nu }\left(z\right)$: cylinder function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $k$: nonnegative integer and $\nu$: complex parameter

A&S Ref:

9.1.80

Referenced by:

§10.23(ii), §10.23(ii), §10.23(ii), §10.44(ii), §10.60(i), §10.60(ii)

Permalink:

http://dlmf.nist.gov/10.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

►where $C_k^{(\nu )}\left(\cos \alpha \right)$ is Gegenbauer’s polynomial (§18.3). … ►

10.23.9 $${\mathrm{e}}^{\mathrm{i}v\cos \alpha }=\frac{\mathrm{\Gamma }\left(\nu \right)}{{(\frac{1}{2}v)}^{\nu }}\sum _{k=0}^{\mathrm{\infty }}(\nu +k){\mathrm{i}}^k{J}_{\nu +k}\left(v\right)C_k^{(\nu )}\left(\cos \alpha \right),$$ $\nu \ne 0,-1,\mathrm{\dots }$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $k$: nonnegative integer and $\nu$: complex parameter

A&S Ref:

9.1.81

Referenced by:

§10.23(ii)

Permalink:

http://dlmf.nist.gov/10.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

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14: 18.3 Definitions

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, 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
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Ultraspherical (Gegenbauer)	$C_n^{(\lambda )}\left(x\right)$	$(-1,1)$	${(1-x^2)}^{\lambda -\frac{1}{2}}$	${\displaystyle \frac{2^{1-2\lambda }\pi \mathrm{\Gamma }\left(n+2\lambda \right)}{(n+\lambda ){\left(\mathrm{\Gamma }\left(\lambda \right)\right)}^{2}n!}}$	${\displaystyle \frac{2^n{\left(\lambda \right)}_n}{n!}}$	$0$	$\lambda >-\frac{1}{2},\lambda \ne 0$
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15: 15.9 Relations to Other Functions

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Gegenbauer (or Ultraspherical)

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15.9.2 $$C_n^{(\lambda )}\left(x\right)=\frac{{\left(2\lambda \right)}_n}{n!}F(\genfrac{}{}{0pt}{}{-n,n+2\lambda }{\lambda +\frac{1}{2}};\frac{1-x}{2}).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ 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, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

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15.9.3 $$C_n^{(\lambda )}\left(x\right)={(2x)}^n\frac{{\left(\lambda \right)}_n}{n!}F(\genfrac{}{}{0pt}{}{-\frac{1}{2}n,\frac{1}{2}(1-n)}{1-\lambda -n};\frac{1}{x^2}).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ 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, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

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15.9.4 $$C_n^{(\lambda )}\left(\cos \theta \right)={\mathrm{e}}^{n\mathrm{i}\theta }\frac{{\left(\lambda \right)}_n}{n!}F(\genfrac{}{}{0pt}{}{-n,\lambda }{1-\lambda -n};{\mathrm{e}}^{-2\mathrm{i}\theta }).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ 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 and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

… ►This is a generalization of Gegenbauer (or ultraspherical) polynomials (§18.3). …

16: 18.11 Relations to Other Functions

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18.11.1 $${𝖯}_n^m\left(x\right)={\left(\frac{1}{2}\right)}_m{(-2)}^m{(1-x^2)}^{\frac{1}{2}m}{C}_{n-m}^{(m+\frac{1}{2})}\left(x\right)={\left(n+1\right)}_m{(-2)}^{-m}{(1-x^2)}^{\frac{1}{2}m}{P}_{n-m}^{(m,m)}\left(x\right),$$ $0\le m\le n$.

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Symbols:

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

Referenced by:

§14.3(iv), §18.11(i)

Permalink:

http://dlmf.nist.gov/18.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

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17: 18.35 Pollaczek Polynomials

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18.35.8 $$P_n^{(\lambda )}(x;0,0)=C_n^{(\lambda )}\left(x\right),$$

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Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Proved:

Ismail (2009, (5.4.3))(proved)

Referenced by:

§18.30(viii), §18.35(i)

Permalink:

http://dlmf.nist.gov/18.35.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(iii), §18.35 and Ch.18

… ►For the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$, the Meixner–Pollaczek polynomials $P_n^{(\lambda )}(x;\varphi )$ and the associated Meixner–Pollaczek polynomials ${𝒫}_n^{\lambda }(x;\varphi ,c)$ see §§18.3, 18.19 and 18.30(v), respectively. …

18: 18.15 Asymptotic Approximations

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18.15.10 $$C_n^{(\lambda )}\left(\cos \theta \right)=\frac{2^{2\lambda }\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\lambda +1\right)}\frac{{\left(2\lambda \right)}_n}{{\left(\lambda +1\right)}_n}\left(\sum _{m=0}^{M-1}\frac{{\left(\lambda \right)}_m{\left(1-\lambda \right)}_m}{m!{\left(n+\lambda +1\right)}_m}\frac{\cos {\theta }_{n,m}}{{(2\sin \theta )}^{m+\lambda }}+O\left(\frac{1}{n^M}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: 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, $!$: factorial (as in $n!$), $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $m$: nonnegative integer, $n$: nonnegative integer and ${\theta }_{n,m}$

Referenced by:

§18.15(ii), §18.15(iii), §18.15(iii)

Permalink:

http://dlmf.nist.gov/18.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(ii), §18.15 and Ch.18

… ►Asymptotic expansions for $C_n^{(\lambda )}\left(\cos \theta \right)$ can be obtained from the results given in §18.15(i) by setting $\alpha =\beta =\lambda -\frac{1}{2}$ and referring to (18.7.1). …

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

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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

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

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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, $n$: 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

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10.60.8 $${\mathrm{e}}^{z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗂}_n^{(1)}\left(z\right)P_n\left(\cos \alpha \right),$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

A&S Ref:

10.2.36

Permalink:

http://dlmf.nist.gov/10.60.E8

Encodings:

pMML, png, TeX

See also:

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

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20: 3.5 Quadrature

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Table 3.5.17_5: Recurrence coefficients in (3.5.30) and (3.5.30_5) for monic versions $p_n(x)$ and orthonormal versions $q_n(x)$ of the classical orthogonal polynomials.

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$p_n(x)$	$q_n(x)$	${\alpha }_n$	${\beta }_n$	$h_0$
$\frac{1}{k_n}{C}_n^{(\lambda )}\left(x\right)$	$\frac{1}{\sqrt{h_n}}{C}_n^{(\lambda )}\left(x\right)$	$0$	$\frac{n(n+2\lambda -1)}{4(n+\lambda )(n+\lambda -1)}$	$\frac{\sqrt{\pi }\mathrm{\Gamma }(\lambda +\frac{1}{2})}{\mathrm{\Gamma }(\lambda +1)}$
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