Tau-function constructions of the recurrence coefficients of orthogonal polynomials
In this interesting paper, the authors study the asymptotic behaviour of recurrence coefficients of orthogonal polynomials, using ideas and methods from mathematical physics. Recall that if \(w\) is a nonnegative function on the real line, that is positive on a set of positive measure, and for which all moments \(\int x^jw(x) dx\), \(j= 0,1,2,\dots\), are finite, then there is an associated system of orthonormal polynomials \(\{p_n\}^\infty_{n= 0}\), satisfying \[ \int p_n p_mw= \delta_{mn}. \] There is a three term recurrence relation \[ xp_n(x)= a_{n+ 1}p_{n+ 1}(x)+ b_n p_n(x)+ a_np_{n- 1}(x), \] where \(a_n> 0\), \(b_n\in\mathbb{R}\). The authors show how ideas from mathematical physics, involving Coulomb fluid methods, chemical potentials, \(\tau\)-functions, and other ideas, may be used to compute the behaviour of \(a_n\) as \(n\to\infty\). A key role is played by a relation which shows that the \(a_n\) may be expressed as the second derivative of a certain free energy. It is remarkable how some of the tools developed in mathematical physics, bear such a close relationship to the weighted potential theory, that has had a dramatic impact on the analysis of orthogonal polynomials in the last twenty years. Three examples are given: Firstly, Freud weights such as \(w(x)= \exp(-|x|^\alpha)\), \(\alpha> 1\), for which asymptotics for \(a_n\) are well known. Secondly, the authors consider the Erdős weight \(w(x)= \exp(-\exp(|x|^\alpha))\), and show that \(a_n= {B^2_n\over 4}(1+ O(B^{-\alpha}_n))\), where \(B_n\) is the root of the equation \(n=\sqrt{\alpha B_n^\alpha\over 2\pi}\exp(B_n^\alpha)\). This is the first time such precise asymptotics have been given for \(a_n\) for these weights. The third example is the weak exponential weight \(\exp(-c|\ln x|^\alpha)\). In all cases the authors show how the Birkhoff-Trjitzinsky method can then be used to derive asymptotics of orthonormal polynomials.
asymptotic behaviour of recurrence coefficients
scientific article; zbMATH DE number 1221618
scientific article