Biorthogonal polynomials and their properties. Direct and inverse statements - Makarov, V. (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 510This paper is dedicated to the centenary of Academician of the National Academy of Sciences of Ukraine V. S. Korolyuk. We consider a class of biorthogonal polynomials generated by a completely positive bilinear functional with Cauchy-type kernel (direct formulation). For these polynomials, we obtain four-term recurrence relations and differential equations of the minimal order (third and fourth). Moreover, we analyze many other properties of these polynomials. In the inverse statememt, we study the problem of existence of a bilinear functional for which two given sequences of polynomials form a biorthogonal system.