Gegenbauer function

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1: 14.3 Definitions and Hypergeometric Representations
§14.3(iv) Relations to Other Functions
In terms of the Gegenbauer function $C^{(\beta)}_{\alpha}\left(x\right)$ and the Jacobi function $\phi^{(\alpha,\beta)}_{\lambda}\left(t\right)$ (§§15.9(iii), 15.9(ii)):
14.3.21 $\mathsf{P}^{\mu}_{\nu}\left(x\right)=\frac{2^{\mu}\Gamma\left(1-2\mu\right)% \Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)\Gamma\left(1-\mu% \right)\left(1-x^{2}\right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{\nu+\mu}\left(x% \right).$
14.3.22 $P^{\mu}_{\nu}\left(x\right)=\frac{2^{\mu}\Gamma\left(1-2\mu\right)\Gamma\left(% \nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)\Gamma\left(1-\mu\right)\left(x^% {2}-1\right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{\nu+\mu}\left(x\right).$
2: 15.9 Relations to Other Functions
Gegenbauer (or Ultraspherical)
15.9.2 $C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}F\left({% -n,n+2\lambda\atop\lambda+\frac{1}{2}};\frac{1-x}{2}\right).$
15.9.4 $C^{(\lambda)}_{n}\left(\cos\theta\right)={\mathrm{e}}^{n\mathrm{i}\theta}\frac% {{\left(\lambda\right)_{n}}}{n!}F\left({-n,\lambda\atop 1-\lambda-n};{\mathrm{% e}}^{-2\mathrm{i}\theta}\right).$
§15.9(iii) GegenbauerFunction
15.9.15 $C^{(\lambda)}_{\alpha}\left(z\right)=\frac{\Gamma\left(\alpha+2\lambda\right)}% {\Gamma\left(2\lambda\right)\Gamma\left(\alpha+1\right)}F\left({-\alpha,\alpha% +2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-z}{2}\right).$
4: 18.10 Integral Representations
18.10.1 $\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma% \left(\alpha+1\right)}{{\pi}^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right% )}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{% 2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\,\mathrm{d}\phi,$ $0<\theta<\pi$, $\alpha>-\tfrac{1}{2}$.
18.10.4 ${\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}}=\frac{\Gamma\left(\alpha+1\right)}{{% \pi}^{\frac{1}{2}}\Gamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos\theta+i% \sin\theta\cos\phi)^{n}\*(\sin\phi)^{2\alpha}\,\mathrm{d}\phi},$ $\alpha>-\frac{1}{2}$.
5: Errata
• The Gegenbauer function $C^{(\lambda)}_{\alpha}\left(z\right)$, was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial $C^{(\lambda)}_{n}\left(z\right)$. 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).

• 6: Bibliography D
• L. Durand (1975) 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.
• 7: 18.12 Generating Functions
18.12.6 $\Gamma\left(\lambda+\tfrac{1}{2}\right){\mathrm{e}}^{z\cos\theta}(\tfrac{1}{2}% z\sin\theta)^{\frac{1}{2}-\lambda}J_{\lambda-\frac{1}{2}}\left(z\sin\theta% \right)=\sum_{n=0}^{\infty}\frac{C^{(\lambda)}_{n}\left(\cos\theta\right)}{{% \left(2\lambda\right)_{n}}}z^{n},$ $0\leq\theta\leq\pi$.
8: 18.15 Asymptotic Approximations
18.15.10 $C^{(\lambda)}_{n}\left(\cos\theta\right)=\frac{2^{2\lambda}\Gamma\left(\lambda% +\frac{1}{2}\right)}{{\pi}^{\frac{1}{2}}\Gamma\left(\lambda+1\right)}\frac{{% \left(2\lambda\right)_{n}}}{{\left(\lambda+1\right)_{n}}}\*\left(\sum_{m=0}^{M% -1}\dfrac{{\left(\lambda\right)_{m}}{\left(1-\lambda\right)_{m}}}{m!\,{\left(n% +\lambda+1\right)_{m}}}\dfrac{\cos\theta_{n,m}}{(2\sin\theta)^{m+\lambda}}+O% \left(\frac{1}{n^{M}}\right)\right),$
9: 18.14 Inequalities
18.14.7 ${(n+\lambda)^{1-\lambda}(1-x^{2})^{\frac{1}{2}\lambda}|C^{(\lambda)}_{n}\left(% x\right)|<\frac{2^{1-\lambda}}{\Gamma\left(\lambda\right)}},$ $-1\leq x\leq 1$, $0<\lambda<1$.
10: 18.17 Integrals
18.17.5 $\frac{C^{(\lambda)}_{n}\left(\cos\theta_{1}\right)}{C^{(\lambda)}_{n}\left(1% \right)}\frac{C^{(\lambda)}_{n}\left(\cos\theta_{2}\right)}{C^{(\lambda)}_{n}% \left(1\right)}=\frac{\Gamma\left(\lambda+\frac{1}{2}\right)}{{\pi}^{\frac{1}{% 2}}\Gamma\left(\lambda\right)}\*\int_{0}^{\pi}\frac{C^{(\lambda)}_{n}\left(% \cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\cos\phi\right)}{C^{(% \lambda)}_{n}\left(1\right)}(\sin\phi)^{2\lambda-1}\,\mathrm{d}\phi,$ $\lambda>0$.
18.17.12 $\frac{\Gamma\left(\lambda-\mu\right)C^{(\lambda-\mu)}_{n}\left(x^{-\frac{1}{2}% }\right)}{x^{\lambda-\mu+\frac{1}{2}n}}=\int_{x}^{\infty}\frac{\Gamma\left(% \lambda\right)C^{(\lambda)}_{n}\left(y^{-\frac{1}{2}}\right)}{y^{\lambda+\frac% {1}{2}n}}\frac{(y-x)^{\mu-1}}{\Gamma\left(\mu\right)}\,\mathrm{d}y,$ $\lambda>\mu>0$, $x>0$,
18.17.13 $\frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{2}}}{\Gamma\left(\lambda+\mu% +\tfrac{1}{2}\right)}\frac{C^{(\lambda+\mu)}_{n}\left(x^{-\frac{1}{2}}\right)}% {C^{(\lambda+\mu)}_{n}\left(1\right)}=\int_{1}^{x}\frac{y^{\frac{1}{2}n}(y-1)^% {\lambda-\frac{1}{2}}}{\Gamma\left(\lambda+\tfrac{1}{2}\right)}\frac{C^{(% \lambda)}_{n}\left(y^{-\frac{1}{2}}\right)}{C^{(\lambda)}_{n}\left(1\right)}% \frac{(x-y)^{\mu-1}}{\Gamma\left(\mu\right)}\,\mathrm{d}y,$ $\mu>0$, $x>1$.
18.17.17 $\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{2n}\left(x\right)% \cos\left(xy\right)\,\mathrm{d}x=\frac{(-1)^{n}\pi\Gamma\left(2n+2\lambda% \right)J_{\lambda+2n}\left(y\right)}{(2n)!\Gamma\left(\lambda\right)(2y)^{% \lambda}},$
18.17.18 $\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{2n+1}\left(x\right)% \sin\left(xy\right)\,\mathrm{d}x=\frac{(-1)^{n}\pi\Gamma\left(2n+2\lambda+1% \right)J_{2n+\lambda+1}\left(y\right)}{(2n+1)!\Gamma\left(\lambda\right)(2y)^{% \lambda}}.$