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 $\exp_{_{\{{\scriptstyle κ}\}}}(x)= (\sqrt{1+κ^2x^2}+κx)^{1/κ}; \exp_{_{\{{\scriptstyle 0}\}}}(x)=\exp (x)$, which presents a power law asymptotic behaviour, has been proposed. The statistical distribution $f=Z^{-1}\exp_{_{\{{\scriptstyle κ}\}}}[-β(E-μ)]$, 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 $κ$-algebra and after introducing the $κ$-analysis, we obtain the $κ$-exponential $\exp_{_{\{{\scriptstyle κ}\}}}(x)$ as the eigenstate of the $κ$-derivative and study its main mathematical properties.