From Taylor to quadratic Hermite-Pade polynomials

Taylor polynomials, Pade approximants, and algebraic Hermite-Pade approximants form a hierarchy of approximation concepts of growing complexity. In the present contribution we climb this ladder of concepts by reviewing results about the asymptotic behaviour of polynomials that are connected with the three concepts. In each case the concepts are used for the approximation of the exponential function. The review starts with a classical result by G. Szego about the asymptotic behaviour of zeros of the Taylor polynomials, it is then continued with asymptotic results by E. B. Saff and R. S. Varga about the asymptotic behaviour of zeros and poles of Pade approximants, and in the last part, analogous results are considered with respect to quadatic Hermite-Pade polynomials. Here, known results are reviewed and some new ones are added. The new results are concerned with the non-diagonal case of quadatic Hermite-Pade polynomials.