ON TRANSFORMATIONS AND ZEROS OF POLYNOMIALS

We survey certain transformations of the set pi-n[x] of n-th degree polynomials into themselves. These transformations share the property that polynomials with all their zeros in a certain real interval are mapped to polynomials with all their zeros in another real interval. Rich sources of such "zero-mapping" transformations can be found in the Laguerre-Polya-Schur theory of multiplier sequences. Others follow from the theory of biorthogonal polynomials, by identifying them with a mapping from the parameter space. This identification leads to two general techniques for the generation of such transformations. As a consequence, we prove a result on the zeros of certain convolution orthogonal polynomials introduced by Al-Salam and Ismail.