# Jacobi polynomials

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##### 1: 18.3 Definitions
###### §18.3 Definitions
This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. 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). … However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6). …
##### 2: 18.14 Inequalities
###### Jacobi
Let $R_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)/P^{(\alpha,\beta)}_{n}\left(1\right)$. …
##### 3: 18.7 Interrelations and Limit Relations
18.7.13 $\frac{P^{(\alpha,\alpha)}_{2n}\left(x\right)}{P^{(\alpha,\alpha)}_{2n}\left(1% \right)}=\frac{P^{(\alpha,-\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(\alpha,% -\frac{1}{2})}_{n}\left(1\right)},$
18.7.14 $\frac{P^{(\alpha,\alpha)}_{2n+1}\left(x\right)}{P^{(\alpha,\alpha)}_{2n+1}% \left(1\right)}=\frac{xP^{(\alpha,\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(% \alpha,\frac{1}{2})}_{n}\left(1\right)}.$
##### 4: 18.6 Symmetry, Special Values, and Limits to Monomials
###### §18.6(ii) Limits to Monomials
18.6.2 $\lim_{\alpha\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,% \beta)}_{n}\left(1\right)}=\left(\frac{1+x}{2}\right)^{n},$
18.6.3 $\lim_{\beta\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,% \beta)}_{n}\left(-1\right)}=\left(\frac{1-x}{2}\right)^{n},$
##### 5: 18.9 Recurrence Relations and Derivatives
For $p_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)$, …
18.9.3 $P^{(\alpha,\beta-1)}_{n}\left(x\right)-P^{(\alpha-1,\beta)}_{n}\left(x\right)=% P^{(\alpha,\beta)}_{n-1}\left(x\right),$
18.9.4 $(1-x)P^{(\alpha+1,\beta)}_{n}\left(x\right)+(1+x)P^{(\alpha,\beta+1)}_{n}\left% (x\right)=2P^{(\alpha,\beta)}_{n}\left(x\right).$
###### Jacobi
18.9.17 $(2n+\alpha+\beta)(1-x^{2})\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{n}% \left(x\right)=n\left(\alpha-\beta-(2n+\alpha+\beta)x\right)P^{(\alpha,\beta)}% _{n}\left(x\right)+2(n+\alpha)(n+\beta)P^{(\alpha,\beta)}_{n-1}\left(x\right),$
##### 6: 18.30 Associated OP’s
###### Associated JacobiPolynomials
18.30.4 $P^{(\alpha,\beta)}_{n}\left(x;c\right)=p_{n}(x;c),$ $n=0,1,\dots$,
18.30.5 $\frac{(-1)^{n}{\left(\alpha+\beta+c+1\right)_{n}}n!\,P^{(\alpha,\beta)}_{n}% \left(x;c\right)}{{\left(\alpha+\beta+2c+1\right)_{n}}{\left(\beta+c+1\right)_% {n}}}=\sum_{\ell=0}^{n}\frac{{\left(-n\right)_{\ell}}{\left(n+\alpha+\beta+2c+% 1\right)_{\ell}}}{{\left(c+1\right)_{\ell}}{\left(\beta+c+1\right)_{\ell}}}% \left(\tfrac{1}{2}x+\tfrac{1}{2}\right)^{\ell}\*{{}_{4}F_{3}}\left({\ell-n,n+% \ell+\alpha+\beta+2c+1,\beta+c,c\atop\beta+\ell+c+1,\ell+c+1,\alpha+\beta+2c};% 1\right),$
For corresponding corecursive associated Jacobi polynomials see Letessier (1995). …
18.30.6 $P_{n}\left(x;c\right)=P^{(0,0)}_{n}\left(x;c\right),$ $n=0,1,\dots$.
##### 7: 18.4 Graphics Figure 18.4.1: Jacobi polynomials P n ( 1.5 , - 0.5 ) ⁡ ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify Figure 18.4.2: Jacobi polynomials P n ( 1.25 , 0.75 ) ⁡ ( x ) , n = 7 , 8 . This illustrates inequalities for extrema of a Jacobi polynomial; see (18.14.16). … Magnify
##### 9: 18.10 Integral Representations
18.10.1 $\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma% \left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right)}% (\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{2}% )\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\mathrm{d}\phi,$ $0<\theta<\pi$, $\alpha>-\tfrac{1}{2}$.
###### Jacobi
18.10.3 $\frac{P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\beta)}_{n}% \left(1\right)}=\frac{2\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma% \left(\alpha-\beta\right)\Gamma\left(\beta+\tfrac{1}{2}\right)}\*\int_{0}^{1}% \int_{0}^{\pi}\left((\cos\tfrac{1}{2}\theta)^{2}-r^{2}(\sin\tfrac{1}{2}\theta)% ^{2}+ir\sin\theta\cos\phi\right)^{n}(1-r^{2})^{\alpha-\beta-1}r^{2\beta+1}(% \sin\phi)^{2\beta}\mathrm{d}\phi\mathrm{d}r,$ $\alpha>\beta>-\tfrac{1}{2}$.
18.10.4 ${\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}}=\frac{\Gamma\left(\alpha+1\right)}{% \pi^{\frac{1}{2}}\Gamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos\theta+i% \sin\theta\cos\phi)^{n}\*(\sin\phi)^{2\alpha}\mathrm{d}\phi},$ $\alpha>-\frac{1}{2}$.
##### 10: 18.12 Generating Functions
###### Jacobi
18.12.1 $\frac{2^{\alpha+\beta}}{R(1+R-z)^{\alpha}(1+R+z)^{\beta}}=\sum_{n=0}^{\infty}P% ^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$ $R=\sqrt{1-2xz+z^{2}}$, $|z|<1$.
18.12.2 $\left(\tfrac{1}{2}(1-x)z\right)^{-\frac{1}{2}\alpha}J_{\alpha}\left(\sqrt{2(1-% x)z}\right)\*\left(\tfrac{1}{2}(1+x)z\right)^{-\frac{1}{2}\beta}I_{\beta}\left% (\sqrt{2(1+x)z}\right)=\sum_{n=0}^{\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x% \right)}{\Gamma\left(n+\alpha+1\right)\Gamma\left(n+\beta+1\right)}z^{n}.$
18.12.3 $(1+z)^{-\alpha-\beta-1}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}(\alpha+\beta+1),% \tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2(x+1)z}{(1+z)^{2}}\right)=% \sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right)_{n}}}{{\left(\beta+1% \right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$ $|z|<1$,
18.12.4 $(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{(\lambda)}_{n}\left(x\right)z^{% n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\tfrac% {1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x% \right)z^{n},$ $|z|<1$.