Les derivees partielles des fonctions pseudo-booleennes generalisees

This paper is devoted to a study of differential calculus for generalised pseudo-Boolean functions with finite domain and antidomain P with a ring structure which may be infinite. A completely new definition is given of partial derivatives of generalised pseudo-Boolean functions $\text{?f}{}_{\mathrm{q}}\text{?x}{}_{\mathrm{i}}\text{=f(x}{}_1\text{,…,x}{}_{\mathrm{i?1}}\text{,a,x}{}_{\mathrm{i+1}}\text{,…,x}{}_{\mathrm{n}}\text{)?f(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)}$. Further are given some properties of these partial derivatives and a new Taylor expansion.M. Davio, J.P. Deschamps, A. Thayse, Belgian mathematicians, have studied the differential calculus for discrete functions with finite domain and antidomain which is a totally ordered lattice 0<1<?<m?1 with a structure of a ring of integers modulo m. If P is finite, then generalised pseudo-Boolean functions are discrete functions, but the partial derivatives of generalised pseudo-Boolean functions are different from the partial derivatives of discrete functions which were studied by the Belgian mathematicians.