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

The family of general Jacobi polynomials Pn(α,β) where α,β∈C can be characterised by complex (non-Hermitian) orthogonality relations (cf. Kuijlaars et al. (2005)). The special subclass of Jacobi polynomials Pn(α,β) 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 Pn(α,β) with α,β∈C, β=α¯ which are known as Pseudo-Jacobi polynomials. The sequence of Pseudo-Jacobi polynomials {Pnα,α¯}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 quasi-orthogonality properties and study the zeros of Pseudo-Jacobi polynomials, providing asymptotics, bounds and results on the monotonicity and convexity of the zeros.