Kaniadakis statistics (also known as κ-statistics) is a generalization of Boltzmann–Gibbs statistical mechanics, based on a relativistic generalization of the classical Boltzmann–Gibbs–Shannon entropy (commonly referred to as Kaniadakis entropy or κ-entropy). Introduced by the Greek Italian physicist Giorgio Kaniadakis in 2001, κ-statistical mechanics preserve the main features of ordinary statistical mechanics and have attracted the interest of many researchers in recent years. The κ-distribution is currently considered one of the most viable candidates for explaining complex physical, natural or artificial systems involving power-law tailed statistical distributions. Kaniadakis statistics have been adopted successfully in the description of a variety of systems in the fields of cosmology, astrophysics, condensed matter, quantum physics, seismology, genomics, economics, epidemiology, and many others.

The mathematical formalism of κ-statistics is generated by κ-deformed functions, especially the κ-exponential function.

The Kaniadakis exponential (or κ-exponential) function is a one-parameter generalization of an exponential function, given by:

with ${\displaystyle \exp _{-\kappa }(x)=\exp _{\kappa }(x)}$.

The κ-exponential for ${\displaystyle 0<\kappa <1}$ can also be written in the form:

The first five terms of the Taylor expansion of ${\displaystyle \exp _{\kappa }(x)}$ are given by:

${\displaystyle \exp _{\kappa }(x)=1+x+{\frac {x^{2}}{2}}+(1-\kappa ^{2}){\frac {x^{3}}{3!}}+(1-4\kappa ^{2}){\frac {x^{4}}{4!}}+\cdots }$

where the first three are the same as a typical exponential function.