Biorthogonal polynomials: Recent developments

An important application of biorthogonal polynomials is in the generation of polynomial transformations that map zeros in a predictable way. This requires the knowledge of the explicit form of the underlying biorthogonal polynomials. The most substantive set of parametrized Bowl measures whose biorthogonal polynomials are known explicitly are the Mobius quotient functions (MQFs), whose moments are Mobius functions in the parameter. In this paper we describe recent work on the characterization of MQFs, following two distinct approaches. Firstly, by restricting the attention to specific families of Borel measures, of the kind that featured in [4], it is possible sometimes to identify all possible MQFs by identifying a functional relationship between weight functions for different values of the parameter. Secondly, provided that the coefficients in Mobius functions are smooth (in a well defined sense), it is possible to prove that the weight function obeys a differential relationship that, in specific cases, allows an explicit characterization of MQFs. In particular, if all such coefficients are polynomial, the MQFs form a subset of generalized hypergeometric functions.