BIORTHOGONAL POLYNOMIALS AND ZERO-MAPPING TRANSFORMATIONS

The authors have presented in [1] a technique to generate transformati ons tau of the set P-n of n(th) degree polynomials to itself such that if p is an element of P-n has all its zeros in (c, d) then tau{p} has all its zeros in (a, b), where (a, b) and (c, d) are given real inter vals. The technique rests upon the derivation of an explicit form of b iorthogonal polynomials whose Borel measure is strictly sign consisten t and such that the ratio of consecutive generalized moments is a rati onal [1/1] function of the parameter. Specific instances of strictly s ign consistent measures that have been debated in [1] include x(mu)d p si(x), mu(x)d psi(x) and x(logq mu)d psi(x), q is an element of (0,1). In this paper, we identify all measures psi such that their consecuti ve generalized moments have a rational [1/1] quotient, thereby charact erizing all possible zero-mapping transformations of this kind.