Jacobi polynomials

=== Other properties ===

=== Other properties ===

The [[discriminant]] is<ref>{{Cite web |title=DLMF: §18.16 Zeros ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.16 |website=dlmf.nist.gov}}</ref><math display="block">\operatorname{Disc}\left(P_n^{(\alpha, \beta)}\right)=2^{-n(n-1)} \prod_{j=1}^n j^{j-2 n+2}(j+\alpha)^{j-1}(j+\beta)^{j-1}(n+j+\alpha+\beta)^{n-j}</math>

The [[discriminant]] is<ref>{{Cite web |title=DLMF: §18.16 Zeros ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.16 |website=dlmf.nist.gov}}</ref><math display="block">\operatorname{Disc}\left(P_n^{(\alpha, \beta)}\right)=2^{-n(n-1)} \prod_{j=1}^n j^{j-2 n+2}(j+\alpha)^{j-1}(j+\beta)^{j-1}(n+j+\alpha+\beta)^{n-j}</math>

== Zeroes ==

If <math>\alpha, \beta > -1</math>, then <math>P_n^{(\alpha, \beta)}</math> has <math>n</math> real roots. Thus in this section we assume <math>\alpha, \beta > -1</math> by default.

The zeroes satisfy the '''[[Thomas Joannes Stieltjes|Stieltjes]] relations''':<ref name=":0">{{Cite journal |last=Marcellán |first=F. |last2=Martínez-Finkelshtein |first2=A. |last3=Martínez-González |first3=P. |date=2007-10-15 |title=Electrostatic models for zeros of polynomials: Old, new, and some open problems |url=https://www.sciencedirect.com/science/article/pii/S037704270600611X |journal=Journal of Computational and Applied Mathematics |series=Proceedings of The Conference in Honour of Dr. Nico Temme on the Occasion of his 65th birthday |volume=207 |issue=2 |pages=258–272 |doi=10.1016/j.cam.2006.10.020 |issn=0377-0427}}</ref><ref>{{Harvard citation|Szegő|1975|p=|loc=Section 6.7. Electrostatic interpretation of the zeros of the classical polynomials}}</ref><math display="block">\begin{aligned}

\sum_{1 \leq j \leq n, i \neq j} \frac{1}{x_{i} - x_{j}} &= \frac{1}{2}\left(\frac{\alpha+1}{1-x_i}-\frac{\beta+1}{1+x_i}\right)\\

\sum_{1 \leq j \leq n} \frac{1}{1 - x_{j}} &= \frac{n(n + \alpha + \beta + 1)}{2(\alpha + 1)} \\

\sum_{1 \leq j \leq n} \frac{1}{1 + x_{j}} &= \frac{n(n + \alpha + \beta + 1)}{2(\beta + 1)} \\

\sum_{1 \leq j \leq n} x_j &= \frac{n (\beta - \alpha)}{2n + \alpha + \beta}

\end{aligned}</math>The first relation can be interpreted physically. Fix an electric particle at -1 with charge <math>\frac{1+\alpha}{2}</math>, and another particle at +1 with charge <math>\frac{1+\beta}{2}</math>. Then, place <math>n </math> electric particles with charge <math>+1 </math>. The first relation states that the zeroes of <math>P_n^{(\alpha, \beta)}</math> are the equilibrium positions of the particles.

Other relations, such as <math>\sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{i} - x_{j})^2}, \sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{i} - x_{j})^3}</math>, are known in closed form.<ref name=":0" />

As the zeroes specify the polynomial up to scaling, this provides an alternative way to uniquely characterize the Jacobi polynomials.

==Asymptotics==

==Asymptotics==