Quasi-orthogonality with applications to some families of classical orthogonal polynomials
Applied Numerical Mathematics
New inequalities from classical Sturm theorems
Journal of Approximation Theory
Full length article: Bounds for extreme zeros of some classical orthogonal polynomials
Journal of Approximation Theory
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The family of general Jacobi polynomials P"n^(^@a^,^@b^) where @a,@b@?C can be characterised by complex (non-Hermitian) orthogonality relations (cf. Kuijlaars et al. (2005)). The special subclass of Jacobi polynomials P"n^(^@a^,^@b^) where @a,@b@?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^(^@a^,^@b^) with @a,@b@?C, @b=@a@? which are known as Pseudo-Jacobi polynomials. The sequence of Pseudo-Jacobi polynomials {P"n^@a^,^@a^@?}"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 @a@?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.