Orthogonality and asymptotics of Pseudo-Jacobi polynomials for non-classical parameters

Abstract

The family of general Jacobi polynomials P(α,β) n where α, β ∈ C can be characterised by complex (non- Hermitian) orthogonality relations (cf. Kuijlaars et al. (2005)). The special subclass of Jacobi polynomials P(α,β) n where α, β ∈ R are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the n real zeros, have been well studied. There is another special subclass of Jacobi polynomials P(α,β) n with α, β ∈ C, β = α which are known as Pseudo-Jacobi polynomials. The sequence of Pseudo-Jacobi polynomials {Pα,α n }∞n =0 is the only other subclass in the general Jacobi family (beside the classical Jacobi polynomials) that has n real zeros for every n = 0, 1, 2, . . . for certain values of α ∈ C. For some parameter ranges Pseudo-Jacobi polynomials are fully orthogonal, for others there is only complex (non-Hermitian) orthogonality.We summarise the orthogonality and quasiorthogonality properties and study the zeros of Pseudo-Jacobi polynomials, providing asymptotics, bounds and results on the monotonicity and convexity of the zeros.