A new one-parameter deformation of the exponential function

Recently, in Kaniadakis (Physica A 296 (2001) 405), a new one-parameter deformation for the exponential function exp{κ}(x)=(1+κ2x2+κx)1/κ; exp{0}(x)=exp(x), which presents a power-law asymptotic behaviour, has been proposed. The statistical distribution f=Z−1exp{κ}[−β(E−μ)], has been obtained both as stable stationary state of a proper nonlinear 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{κ}(x) as the eigenstate of the κ-derivative and study its main mathematical properties.