Deformed Mathematical Objects Stemming from the q-Logarithm Function

Abstract

Generalized numbers, arithmetic operators, and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of q-logarithm/q-exponential inverse functions. Some of the objects were previously described in the literature, while others are newly defined. Commutativity, associativity, and distributivity, and also a pair of linear/nonlinear derivatives, are observed within each class. Two entropic functionals emerge from the formalism, and one of them is the nonadditive Tsallis entropy.

Along the lines of nonadditive entropies: $q$-prime numbers and $q$-zeta functions

“…use a superscript for its algebraic operators, q , a notation that was not adopted in Ref. [8]. The other algebra, corresponding to Eq.…”

“…(1.9), is presented in Section 5, and it corresponds to the oel-arithmetics addressed in Section III.D of Ref. [8], where {q} is here noted q .…”

“…when [1 + (1 − q)z] ≥ 0; otherwise it vanishes. The definitions of the q-logarithm and qexponential functions allow consistent generalizations of algebras [5,6,7,8,9], calculus [6,10,11,8] (see also [12]) and generalized numbers [13,8].…”

“…These four possibilities are explored in Ref. [8] 1 . Other generalizations exist in the literature, also referred to as qnumbers [14,15].…”

“…(1.4), equivalent to the iel-number Eq. (11a) of [8]. For this choice, two algebras will be focused on here, namely,…”