论文标题
Stieltjes - 熟练的问题和退化的正交多项式
The Stieltjes--Fekete problem and degenerate orthogonal polynomials
论文作者
论文摘要
stieltjes的著名结果将经典正交多项式的零与最小化合适能量的线上的点配置有关。能量具有对数相互作用和一个外部场，其指数与经典正交多项式的重量有关。 最佳配置满足了代数方程组：我们将此代数方程式称为stieltjes-《问题问题》，或等效地将stieltjes-extjes-bethe方程式称为。在这项工作中，当外场的导数是任意的有理复杂函数时，我们考虑了stieltjes-fekete问题。我们表明，它的解决方案与某些非纯正正交多项式的零是一对一的对应关系，这些多项式满足过多的正交条件，因此被称为“退化”。这概括了stieltjes的原始结果。
A famous result of Stieltjes relates the zeroes of the classical orthogonal polynomials with the configurations of points on the line that minimize a suitable energy. The energy has logarithmic interactions and an external field whose exponential is related to the weight of the classical orthogonal polynomials. The optimal configuration satisfies an algebraic set of equations: we call this set of algebraic equations the Stieltjes--Fekete problem or equivalently the Stieltjes--Bethe equations. In this work we consider the Stieltjes-Fekete problem when the derivative of the external field is an arbitrary rational complex function. We show that its solutions are in one-to-one correspondence with the zeroes of certain non-hermitean orthogonal polynomials that satisfy an excess of orthogonality conditions and are thus termed "degenerate". This generalizes the original result of Stieltjes.
