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

The family of general Jacobi polynomials P-n((alpha,beta)) where beta epsilon C can be characterised by complex (nonHermitian) orthogonality relations (cf. Kuijlaars et al. (2005)). The special subclass of Jacobi polynomials P-n((alpha,beta)) where alpha, beta epsilon 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-n((alpha,beta)) with alpha, beta epsilon C, beta = (alpha) over bar which are known as Pseudo-Jacobi polynomials. The sequence of Pseudo-Jacobi polynomials {P-n(alpha,(alpha) over bar)}(n=0)(infinity) 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 alpha epsilon 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. (C) 2013 Elsevier Inc. All rights reserved.