Products of monodiffric functions

Abstract

The aim of this paper is to further the theory of monodiffric functions introduced by R. P. Isaacs (A Finite Difference Function Theory, Universidad Nacional Tucuman Revista, vol. 2 (1941), pp. 177-201). Besides developing the concept of monodiffric duality and solving an important boundary-value problem, several new methods of integration are presented. Interrelations among these line integrals are analyzed, and analogs are given for Cauchy's Integral Formula for the Derivative and the method of Integration by Parts. Convolution products of monodiffric functions are derived and analyzed both from the algebraic and from the function-theoretic viewpoint . One of these convolution products is then used to define a product operation for monodiffric functions. It-is shown that the class of functions monodiffric in the first quadrant of the discrete plane forms a commutative integral domain with identity with respect to pointwise addition and the product operation defined. It is also shown that the pseudo-powers of z satisfy the law of exponents with respect to this product operation.