Biorthogonal polynomials and zero-mapping transformations

Abstract

The authors have presented in \cite{IN2} a technique to generate transformations $\cal T$ of the set ${\Bbb P}_n$ of $n$th degree polynomials to itself such that if $p\in{\Bbb P}_n$ has all its zeros in $(c,d)$ then ${\cal T}\{p\}$ has all its zeros in $(a,b)$, where $(a,b)$ and $(c,d)$ are given real intervals. The technique rests upon the derivation of an explicit form of biorthogonal polynomials whose Borel measure is strictly sign consistent and such that the ratio of consecutive generalized moments is a rational $[1/1]$ function of the parameter. Specific instances of strictly sign consistent measures that have been debated in \cite{IN2} include $x^\mu\D\psi(x)$, $\mu^x\D\psi(x)$ and $x^{\log_q\mu}\D\psi(x)$, $q\in(0,1)$. In this paper we identify all measures $\psi$ such that their consecutive generalized moments have a rational $[1/1]$ quotient, thereby characterizing all possible zero-mapping transformations of this kind.