The “classical” Laurent biorthogonal polynomials

An analogue of the Hahn theorem for Laurent biorthogonal polynomials (LBP) Pn(z) is studied. Necessary and sufficient conditions (criterion) for derivatives P̃n(z) = (n + 1)−1P′n+1(z) to be LBP are obtained. This criterion can be written as a linear second-order difference equation for the moments. We find examples of such LBP which are different from the well known “classical” LBP proposed by Hendriksen and Van Rossum. The theory of spectral transformations of the LBP (i.e., analogs of Christoffel and Geronimus transforms) is developed. These transformations are shown to be crucial in study of LBP belonging to the Hahn class.