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Laguerre-Freud’s equations for the recurrence coefficients of semi- classical orthogonal polynomials. (English) Zbl 0802.42022

Monic orthogonal polynomials on the real line satisfy on one hand the orthogonality relations \(\int p_ n(x) p_ m(x) d\mu(x)=0\) whenever \(m\neq n\), and on the other hand a three-term recurrence relation \(p_{n+1}(x)= (x-\beta_ n) p_ n(x)-\gamma_ n p_{n-1}(x)\). In various situations one knows the orthogonality measure \(\mu\) and one needs to find the recurrence coefficients \(\beta_ n\), \(\gamma_ n\) \((n>0)\). For measures of the form \(d\mu(x)= e^{-x^{2m}}dx\) with \(- \infty< x<+\infty\) one can find a nonlinear system of equations for these recurrence coefficients, which for \(m= 1,2,3\) was given by Freud. Earlier, Laguerre obtained the same nonlinear recurrences for weights of the form \(\exp(-Q(x))\), where \(Q\) is a polynomial.
The authors extend the technique of Freud and Laguerre to exponential weight functions on the positive real axis, on the interval \([0,1]\), on two disjoint infinite intervals, and on a finite symmetric interval. They use a formalism for linear functionals which is particularly useful for so-called semi-classical orthogonal polynomials. Note that A. Magnus was able to obtain the Laguerre-Freud equations for generalized Jacobi polynomials [Asymptotics for the simplest generalized Jacobi polynomials recurrence coefficients from Freud’s equations: numerical explorations (to appear)].

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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