DIFFERENT TYPES OF THE DEFORMED EXPONENTIAL FUNCTIONS IN THE STATISTICAL MECHANICS

In recent developments in various sciences, the need is noted to define and use deformed versions of the exponential function. In this paper, the consideration of such functions has two purposes: to have the exponential function as their special case, and, even more, to acquit their inauguration from a mathematical point of view. Starting from well-known Tsallis and Kaniadakis versions, we proposed our own deformed versions and connect them with others. It leads to definition of the deformed numbers and operators. We find their series expansions and derived differential and difference properties. They are important in forming and explaining continuous and discrete models of numerous phenomena in statistical mechanics, thermostatics, information theory, cybernetics, control theory, etc. We illustrate it by analyzing of the different versions of Malthus model in population dynamics. Also, we look back on the well-known law of composed interest in the economy by the deformed exponential function.