Tau-Function Constructions of the Recurrence Coefficients of Orthogonal Polynomials

✍ Scribed by Yang Chen; Mourad E.H Ismail; Walter Van Assche

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 283 KB
• Volume: 20
• Category: Article
• ISSN: 0196-8858
• DOI: 10.1006/aama.1997.0574

✦ Synopsis

In this paper we compute the recurrence coefficients of orthogonal polynomials using -function techniques. It is shown that for polynomials orthogonal with respect to positive weight functions on a noncompact interval, the recurrence coefficient can be expressed as the change in the chemical potential which, for sufficiently large N is the second derivative of the free energy with respect to N, the particle number. We give three examples using this technique: Freud weights, Erdos weights, and weak exponential weights.