Generalised Airy polynomials

Abstract

Abstract We consider properties of semi-classical orthogonal polynomials with respect to the generalised Airy weight ω ( x ; t , λ ) = x λ exp − 1 3 x 3 + t x , x ∈ R + with parameters λ > −1 and t ∈ R . We also investigate the zeros and recurrence coefficients of the polynomials. The generalised sextic Freud weight ω ( x ; t , λ ) = | x | 2 λ + 1 exp − x 6 + t x 2 , x ∈ R arises from a symmetrisation of the generalised Airy weight and we study analogous properties of the polynomials orthogonal with respect to this weight.