Generalised Airy Polynomials

Abstract

We consider properties of semi-classical orthogonal polynomials with respect to the generalised Airy weight \[ω(x;t,λ)=x^λ\exp\left(-\tfrac13x^3+tx\right),\qquad x\in \mathbb{R}^+,\] with parameters $\lambda >-1$ and $t\in \mathbb{R}$. We also investigate the zeros and recurrence coefficients of the polynomials. The generalised sextic Freud weight \[ω(x;t,λ)=|x|^{2λ+1}\exp\left(-x^6+tx^2\right), \qquad x\in \mathbb{R},\] arises from a symmetrisation of the generalised Airy weight and we study analogous properties of the polynomials orthogonal with respect to this weight.

22 pages, 1 figure. arXiv admin note: text overlap with arXiv:2004.00260. J. Phys. A: Math. Theor. (2021)

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Cited by in corpus (3)

• Painlevé IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants
• Generalised higher-order Freud weights
• Symmetric Sextic Freud Weight