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Abstract There is the paper by H. Tietze published in 1905 on differential transcendence of solutions of difference Riccati equations. In this paper, we clarify the essence of Tietze's treatment and make it purely algebraic. As an application, the q -Airy equation is studied. [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]