Abstract :
The functions considered are p-valued functions of n p-valued arguments; they may conveniently be represented by functions over the field Jp of integers modulo some prime p. It is noted that if every function can be uniquely written as a mod-p linear combination (equation 1) then (1) may be thought of equivalently as a canonical form or as a vector-space representation, with the bi forming a basis. This latter interpretation suggests the use of matrix multiplication to transform functions from one canonical form to another. The present paper is devoted to two main topics: 1. A consideration of various canonical forms and their analogies to the Taylor and Maclaurin expansions and the Lagrange interpolation formula of real-variable function theory. 2. A derivation of the matrices relating these forms and of expedient matrix-inversion techniques. The inversion of a pn times pn matrix is reduced, in general, to the inversion of n p times p matrices and in some cases simply to transposition or rotation of the matrix. These simplifications greatly facilitate the evaluation of ´power´ series expansions for all inputs and the generation of power series from function tables.
