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- W2798963697 abstract "We study the differential equation $ - (p(x) y')' + q(x) y' = lambda y,$ where $p(x)$ is a polynomial of degree at most 2 and $q(x)$ is a polynomial of degree at most 1. This includes the classical Jacobi polynomials, Hermite polynomials, Legendre polynomials, Chebychev polynomials and Laguerre polynomials. We provide a general electrostatic interpretation of zeros of such polynomials: the set of real numbers $left{x_1, dots, x_nright}$ satisfies $$ p(x_i) sum_{k = 1 atop k neq i}^{n}{frac{2}{x_k - x_i}} = q(x_i) - p'(x_i) qquad mbox{for all}~ 1leq i leq n$$ if and only if they are zeros of a polynomial solving the differential equation. We also derive a system of ODEs depending on $p(x),q(x)$ whose solutions converge to the zeros of the orthogonal polynomial at an exponential rate." @default.
- W2798963697 created "2018-05-07" @default.
- W2798963697 creator A5054778975 @default.
- W2798963697 date "2018-04-25" @default.
- W2798963697 modified "2023-09-27" @default.
- W2798963697 title "Electrostatic Interpretation of Zeros of Orthogonal Polynomials" @default.
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