Some Families of $q$-Sums and $q$-Products

Taekyun Kim, Dae San Kim, Won Sang Chung, Hyuck In Kwon

In this paper, we introduce two new binary operations, the one called $q$-sum and defined on the set of all real numbers and the other called $q$-product and defined on a subset of real numbers, which have potential importance in the study of $q$-numbers. The set of $q$-numbers of all real numbers, for example, is a field when these operations are restricted to it. Also, we introduce new $q$-exponential and $q$-logarithm and show some relations for them. Finally, we give some remarks on the well-known $q$-gamma, $q$-exponential, and $q$-beta functions.