Three algebraic number systems based on the q-addition with applications

In the spirit of our earlier articles on $q-\omega$ special functions, the purpose of this article is to present many new \(q\)-number systems, which are based on the \(q\)-addition, which was introduced in our previous articles and books. First, we repeat the concept biring, in order to prepare for the introduction of the \(q\)-integers, which extend the \(q\)-natural numbers from our previous book. We formally introduce a \(q\)-logarithm for the \(q\)-exponential function for later use. In order to find \(q\)-analogues of the corresponding formulas for the generating functions and \(q\)-trigonometric functions, we also introduce \(q\)-rational numbers. Then the so-called \(q\)-real numbers \(\mathbb{R}_{\oplus_{q}}\), with a norm, a \(q\)-deformed real line, and with three inequalities, are defined. The purpose of the more general \(q\)-real numbers \(\mathbb{R}_{q}\) is to allow the other \(q\)-addition too. The closely related JHC \(q\)-real numbers \(\mathbb{R}_{\boxplus_{q}}\) have applications to several \(q\)-Euler integrals. This brings us to a vector version of the \(q\)-binomial theorem from a previous paper, which is associated with a special case of the \(q\)-Lauricella function. New \(q\)-trigonometric function formulas are given to show the application of this umbral calculus. Then, some equalities between \(q\)-trigonometric zeros and extreme values are proved. Finally, formulas and graphs for \(q\)-hyperbolic functions are shown