# Jacobi polynomials

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##### 1: 18.3 Definitions
###### §18.3 Definitions
For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). … For $-1-\beta>\alpha>-1$ a finite system of Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$ is orthogonal on $(1,\infty)$ with weight function $w(x)=(x-1)^{\alpha}(x+1)^{\beta}$. For $\nu\in\mathbb{R}$ and $N>-\tfrac{1}{2}$ a finite system of Jacobi polynomials $P^{(-N-1+\mathrm{i}\nu,-N-1-\mathrm{i}\nu)}_{n}\left(\mathrm{i}x\right)$ (called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on $(-\infty,\infty)$ with $w(x)=\left(1+x^{2}\right)^{-N-1}{\mathrm{e}}^{2\nu\operatorname{arctan}x}$. …
##### 2: 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 $\frac{P^{(\alpha,\alpha)}_{2n+1}\left(x\right)}{P^{(\alpha,\alpha)}_{2n+1}% \left(1\right)}=\frac{xP^{(\alpha,\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(% \alpha,\frac{1}{2})}_{n}\left(1\right)}.$
18.7.23 $\lim_{\alpha\to\infty}\alpha^{-\frac{1}{2}n}P^{(\alpha,\alpha)}_{n}\left(% \alpha^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{2^{n}n!}.$
##### 3: 18.14 Inequalities
###### Jacobi
Let $R_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)/P^{(\alpha,\beta)}_{n}\left(1\right)$. …
##### 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 $\lim_{\beta\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,% \beta)}_{n}\left(-1\right)}=\left(\frac{1-x}{2}\right)^{n},$
##### 6: 18.9 Recurrence Relations and Derivatives
For $p_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)$, … 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 $(1-x)P^{(\alpha+1,\beta)}_{n}\left(x\right)+(1+x)P^{(\alpha,\beta+1)}_{n}\left% (x\right)=2P^{(\alpha,\beta)}_{n}\left(x\right),$
##### 7: 18.10 Integral Representations
18.10.1 $\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma% \left(\alpha+1\right)}{{\pi}^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right% )}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{% 2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\,\mathrm{d}\phi,$ $0<\theta<\pi$, $\alpha>-\tfrac{1}{2}$.
Generalizations of (18.10.1) for $P^{(\alpha,\beta)}_{n}$ are given in Gasper (1975, (6),(8)) and Koornwinder (1975a, (5.7),(5.8)). …
###### Jacobi
18.10.4 ${\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}}=\frac{\Gamma\left(\alpha+1\right)}{{% \pi}^{\frac{1}{2}}\Gamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos\theta+i% \sin\theta\cos\phi)^{n}\*(\sin\phi)^{2\alpha}\,\mathrm{d}\phi},$ $\alpha>-\frac{1}{2}$.
##### 8: 18.12 Generating Functions
###### Jacobi
18.12.1 $\frac{2^{\alpha+\beta}}{R(1+R-z)^{\alpha}(1+R+z)^{\beta}}=\sum_{n=0}^{\infty}P% ^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$ $R=\sqrt{1-2xz+z^{2}}$, $|z|<1$,
18.12.3 $(1+z)^{-\alpha-\beta-1}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}(\alpha+\beta+1),% \tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2(x+1)z}{(1+z)^{2}}\right)=% \sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right)_{n}}}{{\left(\beta+1% \right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$ $|z|<1$,
18.12.3_5 $\frac{1+z}{(1-2xz+z^{2})^{\beta+\frac{3}{2}}}=\sum_{n=0}^{\infty}\frac{{\left(% 2\beta+2\right)_{n}}}{{\left(\beta+1\right)_{n}}}P^{(\beta+1,\beta)}_{n}\left(% x\right)z^{n},$ $|z|<1$,
18.12.4 $(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{(\lambda)}_{n}\left(x\right)z^{% n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\tfrac% {1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x% \right)z^{n},$ $|z|<1$.
##### 10: 18.1 Notation
• Jacobi: $P^{(\alpha,\beta)}_{n}\left(x\right)$.

• Big $q$-Jacobi: $P_{n}\left(x;a,b,c;q\right)$.

• Little $q$-Jacobi: $p_{n}\left(x;a,b;q\right)$.

• Nor do we consider the shifted Jacobi polynomials: