Jordaan, Kerstin HeidrunToókos, Ferenc2014-04-012014-04-012014Jordaan, K & Toókos, F 2014, 'Orthogonality and asymptotics of Pseudo-Jacobi polynomials for non-classical parameters', Journal of Approximation Theory, vol.187, pp. 1-12.0021-9045 (print)1096-0430 (online)10.1016/j.jat.2013.10.003http://hdl.handle.net/2263/37328The 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.en© 2013 Elsevier Inc. All rights reserved. All rights reserved. Notice : this is the author’s version of a work that was accepted for publication in Journal of Approximation Theory. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Approximation Theory, vol.178, no.1, pp. 1-12, 2014. doi :10.1016/j.jat.2013.10.003Orthogonal polynomialsQuasi-orthogonal polynomialsJacobi polynomials with complex parametersPseudo-Jacobi polynomialsZerosPostprint Article