### Abstract:

The 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.