Complex orthogonality

The great majority of research contributions in the field of orthogonal polynomials is concerned with concepts that live either on the real axis $\mathbb R$, or in the unit circle $\mathbb T$, and if one goes away from the real axis into the complex plane, like in case of the unit circle, then one usually assumes at least that the orthogonality is based on an Hermitian bilinear form. Orthogonality relations of this type may be written as
\begin{equation} \int \bar z^jp_n(z)\,d\mu(z)=0\quad\text{for}\quad j=0,\dots,n-1, \tag{1} \end{equation}
where $\mu$ is a positive measure with support in $\mathbb C$.
In several areas of rational approximation, like in Padé approximation, rational approximation, and also in the theory of continued fractions, we are confronted with orthogonality relations that no longer are Hermitian. Here, the relations may typically have a form like
\begin{equation} \int_\gamma z^jp_n(z)f(z)\,dz=0\quad\text{for}\quad j=0,\dots,n-1, \tag{2} \end{equation}
with $p_n\in\mathscr P_n$, $\gamma$ an integration path in $\mathbb C$, and $f$ a weight function, which is assumed to be analytic in a domain containing $\gamma$. The crucial difference between the two relations (1) and (2) is the absence of the conjugation in (2), and this little difference has indeed quite dramatic consequences. Orthogonal polynomials $p_n$ defined by (2) have properties very different from those defined by (1). In our talk, we will be concerned with such polynomials.
After the discussion of some examples and some glances at the background of the problem in approximation theory, we will review concepts for the asymptotic analysis of sequences of orthogonal polynomials $p_n$ satisfying (2). We shall compare different strategies for their investigation, and will try to shed light on the critical points in the analysis of the given type of non-Hermitian orthogonality relations.