Polynomials Orthogonal on the Semicircle

Abstract

Orthogonal polynomials throughout the history of scientific works had alternate periods of popularity and weak interest in them – until interest in orthogonal polynomials began to grow rapidly. The reason was the increasing application in different areas. That is the reason that we started to observe polynomials orthogonal on the semicircle introduced by W. Gautschi, H. Landau and G. Milovanović. It turns out that these polynomials, although at first sight completely different because of complex plane, have convenient properties very similar to real orthogonal polynomials. The properties of the polynomials we consider are recurrence relations, location of zeros and the corresponding Gaussian and anti- Gaussian quadrature rules. It had to pass about 20 years before orthogonal polynomials found application in areas as numerical integration, number theory and probability theory and statistics. In the age of technological innovations, one might think that the mentioned applications are no longer significant. However, papers from the last few years show that these polynomials do not lose race with time and show their application with artificial neural networks, in new models of food engineering, as well as they greatly help Markov’s chains in predicting events. That’s why it is very likely that our polynomials orthogonal on the semicircle will also find application in some fields of machine learning, so we don’t want to wait another 20 years to give them a chance.