Gegenbauer function

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1: 14.3 Definitions and Hypergeometric Representations

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§14.3(iv) Relations to Other Functions

►In terms of the Gegenbauer function $C_{\alpha }^{(\beta )}\left(x\right)$ and the Jacobi function ${\varphi }_{\lambda }^{(\alpha ,\beta )}\left(t\right)$ (§§15.9(iii), 15.9(ii)): ►

14.3.21

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $C_{\alpha }^{(\lambda )}\left(z\right)$: Gegenbauer function, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.3(iv), §18.10(i), §18.11(i)

Permalink:

http://dlmf.nist.gov/14.3.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iv), §14.3 and Ch.14

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14.3.22

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $C_{\alpha }^{(\lambda )}\left(z\right)$: Gegenbauer function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.3(iv), §18.11(i)

Permalink:

http://dlmf.nist.gov/14.3.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iv), §14.3 and Ch.14

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

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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

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§15.9(iii) Gegenbauer Function

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15.9.15 $$C_{\alpha }^{(\lambda )}\left(z\right)=\frac{\mathrm{\Gamma }\left(\alpha +2\lambda \right)}{\mathrm{\Gamma }\left(2\lambda \right)\mathrm{\Gamma }\left(\alpha +1\right)}F(\genfrac{}{}{0pt}{}{-\alpha ,\alpha +2\lambda }{\lambda +\frac{1}{2}};\frac{1-z}{2}).$$

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Defines:

$C_{\alpha }^{(\lambda )}\left(z\right)$: Gegenbauer function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.9.E15

Encodings:

pMML, png, TeX

See also:

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

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3: 18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

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$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
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$C_n^{(\lambda )}\left(x\right)$	$1-x^2$	$-(2\lambda +1)x$	$0$	$n(n+2\lambda )$
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4: Errata

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The Gegenbauer function $C_{\alpha }^{(\lambda )}\left(z\right)$, was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial $C_n^{(\lambda )}\left(z\right)$. In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

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5: 18.10 Integral Representations

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

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

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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 $x$ , $\mathrm{i}$: 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

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6: 18.12 Generating Functions

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

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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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7: Bibliography D

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Nicholson-type Integrals for Products of Gegenbauer Functions and Related Topics. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), R. A. Askey (Ed.), pp. 353–374. Math. Res. Center, Univ. Wisconsin, Publ. No. 35.

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External Links:

MathReview (K. M. Saksena), ZentralBlatt

Referenced by:

(12.12.4), §12.12, §18.17(iii), §18.17(iii).

Encodings:

BibTeX

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8: 18.14 Inequalities

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

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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

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9: 18.17 Integrals

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

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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)

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

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

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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

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

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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)

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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18.17.17 $${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_{2n}^{(\lambda )}\left(x\right)\cos \left(xy\right)dx=\frac{{(-1)}^n\pi \mathrm{\Gamma }\left(2n+2\lambda \right)J_{\lambda +2n}\left(y\right)}{(2n)!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }},$$

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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, $\cos z$: cosine function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: 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)

Permalink:

http://dlmf.nist.gov/18.17.E17

Encodings:

pMML, png, TeX

See also:

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

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18.17.18 $${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_{2n+1}^{(\lambda )}\left(x\right)\sin \left(xy\right)dx=\frac{{(-1)}^n\pi \mathrm{\Gamma }\left(2n+2\lambda +1\right)J_{2n+\lambda +1}\left(y\right)}{(2n+1)!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }}.$$

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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 $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v)

Permalink:

http://dlmf.nist.gov/18.17.E18

Encodings:

pMML, png, TeX

See also:

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

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10: 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

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