/sci/ - Science & Math

Let's talk about polynomials of this form for a bit (for some [math] \alpha , \beta \in \mathbb{R} [/math]):

[math] P_n^{(\alpha, \beta)}(x) = \dfrac{1}{2^n} \sum\limits_{j=0}^{n}\binom{n+\alpha}{j}\binom{n+\beta}{n-j}(x-1)^{n-j}(x+1)^j [/math]

It turns out they're eigenfunctions of the (Jacobi) differential operator
[math] y \rightarrow - (1 + x^2)y'' + (\alpha - \beta + (\alpha + \beta + 2)x)y' [/math] with eigenvalue [math] n(n + \alpha + \beta + 1) [/math] and all other eigenfunctions of this operator can be expressed in terms of such polynomials.


(Don't let the bullies discourage you friends.)