About a system of anti-periodic trigonometric functions

In this paper we introduce the so-called second kind trigonometric system, which is a useful tool for the representation of 2π2π “anti-periodic” functions. Using these functions we study a sequence of trigonometric polynomials, which are bi-orthogonal in the Szegő’s sense. We study the usual topics in the theory of Fourier series and we present a new connection with the orthogonal polynomials (OP) on the unit circle and some useful properties like: recurrence relations, kernel representations and a Favard’s type theorem.