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

Abstract: The family of general Jacobi polynomials where can be characterised by complex (non-Hermitian) orthogonality relations (cf. Kuijlaars et al. (2005)). The special subclass of Jacobi polynomials where are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the real zeros, have been well studied. There is another special subclass of Jacobi polynomials with , which are known as Pseudo-Jacobi polynomials. The sequence of Pseudo-Jacobi polynomials is the only other subclass in the general Jacobi family (beside the classical Jacobi polynomials) that has real zeros for every for certain values of . 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. [Copyright &y& Elsevier]