# Gegenbauer polynomials

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##### 1: 18.7 Interrelations and Limit Relations
18.7.1 $C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{{\left(% \lambda+\frac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_% {n}\left(x\right),$
18.7.15 $C^{(\lambda)}_{2n}\left(x\right)=\frac{{\left(\lambda\right)_{n}}}{{\left(% \tfrac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},-\frac{1}{2})}_{n}\left(2x^{2% }-1\right),$
##### 2: 18.9 Recurrence Relations and Derivatives
18.9.7 $(n+\lambda)C^{(\lambda)}_{n}\left(x\right)=\lambda\left(C^{(\lambda+1)}_{n}% \left(x\right)-C^{(\lambda+1)}_{n-2}\left(x\right)\right),$
18.9.8 $4\lambda(n+\lambda+1)(1-x^{2})C^{(\lambda+1)}_{n}\left(x\right)=-(n+1)(n+2)C^{% (\lambda)}_{n+2}\left(x\right)+(n+2\lambda)(n+2\lambda+1)C^{(\lambda)}_{n}% \left(x\right).$
18.9.20 $\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda% )}_{n}\left(x\right)\right)=-\frac{(n+1)(n+2\lambda-1)}{2(\lambda-1)}{(1-x^{2}% )^{\lambda-\frac{3}{2}}}C^{(\lambda-1)}_{n+1}\left(x\right).$
##### 3: 18.1 Notation
• Ultraspherical (or Gegenbauer): $C^{(\lambda)}_{n}\left(x\right)$.

• In Szegő (1975, §4.7) the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$ 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 $C^{(0)}_{n}\left(x\right)=\frac{2}{n}T_{n}\left(x\right)=\frac{2(n-1)!}{{\left% (\tfrac{1}{2}\right)_{n}}}P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(x\right),$ $n=1,2,3,\dots$.
##### 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: 18.12 Generating Functions
18.12.4 $(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{(\lambda)}_{n}\left(x\right)z^{% n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\tfrac% {1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x% \right)z^{n},$ $|z|<1$.
18.12.5 $\frac{1-xz}{(1-2xz+z^{2})^{\lambda+1}}=\sum_{n=0}^{\infty}\frac{n+2\lambda}{2% \lambda}C^{(\lambda)}_{n}\left(x\right)z^{n},$ $|z|<1$.
18.12.6 $\Gamma\left(\lambda+\tfrac{1}{2}\right)e^{z\cos\theta}(\tfrac{1}{2}z\sin\theta% )^{\frac{1}{2}-\lambda}J_{\lambda-\frac{1}{2}}\left(z\sin\theta\right)=\sum_{n% =0}^{\infty}\frac{C^{(\lambda)}_{n}\left(\cos\theta\right)}{{\left(2\lambda% \right)_{n}}}z^{n},$ $0\leq\theta\leq\pi$.
##### 6: 18.6 Symmetry, Special Values, and Limits to Monomials
18.6.4 $\lim_{\lambda\to\infty}\frac{C^{(\lambda)}_{n}\left(x\right)}{C^{(\lambda)}_{n% }\left(1\right)}=x^{n},$
##### 7: 18.14 Inequalities
18.14.4 $|C^{(\lambda)}_{n}\left(x\right)|\leq C^{(\lambda)}_{n}\left(1\right)=\frac{{% \left(2\lambda\right)_{n}}}{n!},$ $-1\leq x\leq 1$, $\lambda>0$.
18.14.5 $|C^{(\lambda)}_{2m}\left(x\right)|\leq|C^{(\lambda)}_{2m}\left(0\right)|=\left% |\frac{{\left(\lambda\right)_{m}}}{m!}\right|,$ $-1\leq x\leq 1$, $-\tfrac{1}{2}<\lambda<0$,
18.14.6 $|C^{(\lambda)}_{2m+1}\left(x\right)|<\frac{-2{\left(\lambda\right)_{m+1}}}{% \left((2m+1)(2\lambda+2m+1)\right)^{\frac{1}{2}}m!},$ $-1\leq x\leq 1$, $-\tfrac{1}{2}<\lambda<0$.
18.14.7 ${(n+\lambda)^{1-\lambda}(1-x^{2})^{\frac{1}{2}\lambda}|C^{(\lambda)}_{n}\left(% x\right)|<\frac{2^{1-\lambda}}{\Gamma\left(\lambda\right)}},$ $-1\leq x\leq 1$, $0<\lambda<1$.
##### 8: 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 $\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{2n+1}\left(x\right)% \sin\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi\Gamma\left(2n+2\lambda+1% \right)J_{2n+\lambda+1}\left(y\right)}{(2n+1)!\Gamma\left(\lambda\right)(2y)^{% \lambda}}.$
##### 9: 18.18 Sums
18.18.16 $C^{(\mu)}_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac% {\lambda+n-2\ell}{\lambda}\frac{{\left(\mu\right)_{n-\ell}}}{{\left(\lambda+1% \right)_{n-\ell}}}\frac{{\left(\mu-\lambda\right)_{\ell}}}{\ell!}C^{(\lambda)}% _{n-2\ell}\left(x\right),$
18.18.17 $(2x)^{n}=n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{\lambda+n-2\ell}% {\lambda}\frac{1}{{\left(\lambda+1\right)_{n-\ell}}\,\ell!}C^{(\lambda)}_{n-2% \ell}\left(x\right).$
18.18.22 $C^{(\lambda)}_{m}\left(x\right)C^{(\lambda)}_{n}\left(x\right)=\sum_{\ell=0}^{% \min(m,n)}\frac{(m+n+\lambda-2\ell)(m+n-2\ell)!}{(m+n+\lambda-\ell)\ell!\,(m-% \ell)!\,(n-\ell)!}\*\frac{{\left(\lambda\right)_{\ell}}{\left(\lambda\right)_{% m-\ell}}{\left(\lambda\right)_{n-\ell}}{\left(2\lambda\right)_{m+n-\ell}}}{{% \left(\lambda\right)_{m+n-\ell}}{\left(2\lambda\right)_{m+n-2\ell}}}C^{(% \lambda)}_{m+n-2\ell}\left(x\right).$
18.18.30 $\sum_{\ell=0}^{n}\frac{\ell+2\lambda}{2\lambda}C^{(\lambda)}_{\ell}\left(x% \right)x^{n-\ell}=C^{(\lambda+1)}_{n}\left(x\right).$
##### 10: 18.3 Definitions
For exact values of the coefficients of the Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$, the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$, 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). …