A new one parameter deformation of the exponential function

Abstract

Recently, in the ref. Physica A \bfm{296} 405 (2001), a new one parameter deformation for the exponential function $\exp_{_{\{{\scriptstyle \kappa}\}}}(x)= (\sqrt{1+\kappa^2x^2}+\kappa x)^{1/\kappa}; \exp_{_{\{{\scriptstyle 0}\}}}(x)=\exp (x)$, which presents a power law asymptotic behaviour, has been proposed. The statistical distribution $f=Z^{-1}\exp_{_{\{{\scriptstyle \kappa}\}}}[-\beta(E-\mu)]$, has been obtained both as stable stationary state of a proper non linear kinetics and as the state which maximizes a new entropic form. In the present contribution, starting from the $\kappa$-algebra and after introducing the $\kappa$-analysis, we obtain the $\kappa$-exponential $\exp_{_{\{{\scriptstyle \kappa}\}}}(x)$ as the eigenstate of the $\kappa$-derivative and study its main mathematical properties.