Generalised Airy Polynomials

We consider properties of semi-classical orthogonal polynomials with respect to the generalised Airy weight $$\omega (x;t,\lambda )=x^{\lambda }\exp (-\frac{1}{3}{x}^3+tx),x\in {\mathbb{R}}^{+},$$ 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 $$\omega (x;t,\lambda )=\mathrm{\mid }x{\mathrm{\mid }}^{2\lambda +1}\exp (-x^6+t{x}^2),x\in \mathbb{R},$$ arises from a symmetrisation of the generalised Airy weight and we study analogous properties of the polynomials orthogonal with respect to this weight.