One and two side generalisations of the log-Normal distribution by means of a new product definition

In this manuscript we introduce a generalisation of the log-Normal distribution that is inspired by a modification of the Kaypten multiplicative process using the $q$-product of Borges [Physica A \textbf{340}, 95 (2004)]. Depending on the value of q the distribution increases the tail for small (when $q<1$) or large (when $q>1$) values of the variable upon analysis. The usual log-Normal distribution is retrieved when $q=1$. The main statistical features of this distribution are presented as well as a related random number generators and tables of quantiles of the Kolmogorov-Smirnov. Lastly, we illustrate the application of this distribution studying the adjustment of a set of variables of biological and financial origin.