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

Ernst, Thomas 

Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 75 (2021) no. 2, pp. 46-71.

Résumé

In the spirit of our earlier articles on q-ω 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 ℝ_⊕_q, with a norm, a q-deformed real line, and with three inequalities, are defined. The purpose of the more general q-real numbers ℝ_q is to allow the other q-addition too. The closely related JHC q-real numbers ℝ_⊞_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.

Détail

Keywords: q-real numbers, q-rational numbers, q-integers, q-trigonometric functions, biring, semiring

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