A new applied approach for executing computations with infinite and infinitesimal quantities

A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework. The new methodology has allowed us to introduce the Infinity Computer working with such numbers (its simulator has already been realized). Examples dealing with divergent series, infinite sets, and limits are given

ALGEBRAIC INDEPENDENCE RESULTS FOR A CERTAIN FAMILY OF POWER SERIES, INFINITE PRODUCTS, AND LAMBERT TYPE SERIES

For a certain class of power series, infinite products, and Lambert type series, we establish a necessary and sufficient condition for the infinite set consisting of their values, as well as their derivatives of any order at any algebraic points except their poles and zeroes, to be algebraically independent. As its corollary, we construct an example of an infinite family of entire functions of two variables with the following property: Their values and their partial derivatives of any order at any distinct algebraic points with nonzero components are algebraically independent