Explicit formulas for five-term recurrence coefficients of orthogonal trigonometric polynomials of semi-integer degree

Orthogonal systems of trigonometric polynomials of semi-integer degree with respect to a weight function w(x) on [0, 2π) have been considered firstly by Turetzkii in [A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (2005) 337–359 (Translation in English from Uchenye Zapiski, Vypusk 1(149), Seria math. Theory of Functions, Collection of Papers, Izdatel’stvo Belgosuniversiteta imeni V.I. Lenina, Minsk (1959) 31–54)]. It is proved that such orthogonal trigonometric polynomials of semi-integer degree satisfy five-term recurrence relation. In this paper we present explicit formulas for five-term recurrence coefficients for some weight functions.