# A new one parameter deformation of the exponential function

Source: [A new one parameter deformation of the exponential function](<https://www.sota2.com/research/paper/68fa31ba1e209131ce8814c8>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T19:02:22.005Z

Authors: G. Kaniadakis, A.M. Scarfone

Published: 2001-09-28T11:47:52.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/cond-mat/0109537>)

[PDF](<https://arxiv.org/pdf/cond-mat/0109537.pdf>)