On the Polynomials Orthogonal on Regular Polygons

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The two-parameter Pastro–Al-Salam–Ismail (PASI) polynomials are known to be bi-orthogonal on the unit circle with continuous weight function when 0<q<1. We study the case ofqa root of unity. It is shown that corresponding PASI polynomials are orthogonal on the unit circle with discrete measure located on the vertices of the regularN-gon. Cases leading to a positive weight function are analyzed. In particular, we obtain trigonometric analogs of the Askey–Szegő polynomials which are orthogonal on regularN-gons with positive weight function.