New bases of monodiffric polynomials

Abstract

Two new bases of monodiffric polynomials are defined. These new bases have interesting properties. Both bases satisfy the definition of Zeilberger's "system of pseudo-powers" for monodiffric polynomials, and the sums Summation a_n pi_n (z) and Summation a_n pi*_n(z) converge absolutely on the upper half-plane and right half-plane, respectively, if, and only if, lim sup |a_n|^(1/n) =0 . It is shown that there is no "system of pseudo-powers", {P_n(z)}, such that Summation a_n P_n(z) converges for every lattice point z, if lim sup |a_n|^(1/n) =0. By using the discrete exponential function corresponding to the basis {pi_k(z)}, it is possible to give discrete analogues for a well-known Paley-Wiener space, the Paley-Wiener theorem, and H^2-spaces. The theory of discrete H^2-spaces analogous to the theory of classical H^2 -spaces is also developed. The final result of this paper establishes a relationship between the class of monodiffric functions and discrete analytic functions. A discrete H^2-space theory is developed for the class of discrete analytic functions on the first quadrant.