A New Basis for Discrete Analytic Polynomials

A new basis {irk(z)}t.o for discrete analytic polynomials is presented for which the series 2k-o ak7Tk(z) converges absolutely to a discrete analytic function in the upper right quarter lattice whenever lim | ak \" k = 0. Introduction Let Z be the group of integers and consider functions / : Z X Z ^ C such that (1.1) f(x, y) + if(x + 1, y) / (* + 1, y + 1) if(x, y + 1) = 0 for every (x, y ) £ Z x Z . Such functions are termed discrete entire. If (1.1) holds only for (x, y)G G, G CZ x Z, then we say that / is discrete analytic in G. Discrete analytic functions were introduced by Ferrand (1944) and the theory was developed by Duffin (1956) and others. Duffin (1956) introduced the following basis for discrete analytic polynomials (z = x + iy), which he called pseudo-powers. Each pk(z) is a discrete entire function and a polynomial of degree k in (x, y). Duffin (1956) showed that every discrete analytic polynomial can be expressed as a linear combination of these pseudo-powers. Duffin and Peterson (1968) introduced an analogue of the McClaurin series in terms of these pseudo-powers. However, their analogue has the dis-