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In this paper, we introduce a way to approximate meromorphic functions belonging to Poincaré's ‘new’ class by rational functions. The class consists of meromorphic functions having a ‘théorèm de multiplication’ with a suitable condition, which is a system of difference equations on the transform sendingto, where. Since graphing rational functions is somewhat easy, we are able to know rough graphs of those ‘new’ functions by our result. Our method uses repeated substitutions of the difference equations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to note that properties of being elementary in differential and difference senses are different as a whole. In this paper, we introduce the notion of finitelyn-reducible difference field extensions and study the irreducibility of algebraic difference equations of order 1. As a result, we find that the exponential functionand the Weierstrass functioncannot be built up by using the variablezand constants, together with repeated algebraic operations and the taking of solutions of linear difference equations on the multiplicationand solutions of such algebraic difference equations as difference Riccati equations on. [ABSTRACT FROM PUBLISHER]