Kaniadakis statistics

Kaniadakis statistics (also known as κ-statistics) is a generalization of Boltzmann–Gibbs statistical mechanics,^[1] based on a relativistic^[2][3][4] 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,^[5] 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,^[6][7] 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,^[8][9][10] astrophysics,^[11][12] condensed matter, quantum physics,^[13][14] seismology,^[15][16] genomics,^[17][18] economics,^[19][20] epidemiology,^[21] 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:

${\displaystyle \exp _{\kappa }(x)={\begin{cases}{\Big (}{\sqrt {1+\kappa ^{2}x^{2}}}+\kappa x{\Big )}^{\frac {1}{\kappa }}&{\text{if }}0<\kappa <1.\\[6pt]\exp(x)&{\text{if }}\kappa =0,\\[8pt]\end{cases}}}$

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

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

${\displaystyle \exp _{\kappa }(x)=\exp {\Bigg (}{\frac {1}{\kappa }}{\text{arcsinh}}(\kappa x){\Bigg )}.}$

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.

Basic properties

The κ-exponential function has the following properties of an exponential function:

${\displaystyle \exp _{\kappa }(x)\in \mathbb {C} ^{\infty }(\mathbb {R} )}$

${\displaystyle {\frac {d}{dx}}\exp _{\kappa }(x)>0}$

${\displaystyle {\frac {d^{2}}{dx^{2}}}\exp _{\kappa }(x)>0}$

${\displaystyle \exp _{\kappa }(-\infty )=0^{+}}$

${\displaystyle \exp _{\kappa }(0)=1}$

${\displaystyle \exp _{\kappa }(+\infty )=+\infty }$

${\displaystyle \exp _{\kappa }(x)\exp _{\kappa }(-x)=-1}$

For a real number ${\displaystyle r}$, the κ-exponential has the property:

${\displaystyle {\Big [}\exp _{\kappa }(x){\Big ]}^{r}=\exp _{\kappa /r}(rx)}$.

The Kaniadakis logarithm (or κ-logarithm) is a relativistic one-parameter generalization of the ordinary logarithm function,

${\displaystyle \ln _{\kappa }(x)={\begin{cases}{\frac {x^{\kappa }-x^{-\kappa }}{2\kappa }}&{\text{if }}0<\kappa <1,\\[8pt]\ln(x)&{\text{if }}\kappa =0,\\[8pt]\end{cases}}}$

with ${\displaystyle \ln _{-\kappa }(x)=\ln _{\kappa }(x)}$, which is the inverse function of the κ-exponential:

${\displaystyle \ln _{\kappa }{\Big (}\exp _{\kappa }(x){\Big )}=\exp _{\kappa }{\Big (}\ln _{\kappa }(x){\Big )}=x.}$

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

${\displaystyle \ln _{\kappa }(x)={\frac {1}{\kappa }}\sinh {\Big (}\kappa \ln(x){\Big )}}$

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

${\displaystyle \ln _{\kappa }(1+x)=x-{\frac {x^{2}}{2}}+\left(1+{\frac {\kappa ^{2}}{2}}\right){\frac {x^{3}}{3}}-\cdots }$

following the rule

${\displaystyle \ln _{\kappa }(1+x)=\sum _{n=1}^{\infty }b_{n}(\kappa )\,(-1)^{n-1}\,{\frac {x^{n}}{n}}}$

with ${\displaystyle b_{1}(\kappa )=1}$, and

${\displaystyle b_{n}(\kappa )(x)={\begin{cases}1&{\text{if }}n=1,\\[8pt]{\frac {1}{2}}{\Big (}1-\kappa {\Big )}{\Big (}1-{\frac {\kappa }{2}}{\Big )}...{\Big (}1-{\frac {\kappa }{n-1}}{\Big )},\,+\,{\frac {1}{2}}{\Big (}1+\kappa {\Big )}{\Big (}1+{\frac {\kappa }{2}}{\Big )}...{\Big (}1+{\frac {\kappa }{n-1}}{\Big )}&{\text{for }}n>1,\\[8pt]\end{cases}}}$

where ${\displaystyle b_{n}(0)=1}$ and ${\displaystyle b_{n}(-\kappa )=b_{n}(\kappa )}$. The two first terms of the Taylor expansion of ${\displaystyle \ln _{\kappa }(x)}$ are the same as an ordinary logarithmic function.

Basic properties

The κ-logarithm function has the following properties of a logarithmic function:

${\displaystyle \ln _{\kappa }(x)\in \mathbb {C} ^{\infty }(\mathbb {R} ^{+})}$

${\displaystyle {\frac {d}{dx}}\ln _{\kappa }(x)>0}$

${\displaystyle {\frac {d^{2}}{dx^{2}}}\ln _{\kappa }(x)<0}$

${\displaystyle \ln _{\kappa }(0^{+})=-\infty }$

${\displaystyle \ln _{\kappa }(1)=0}$

${\displaystyle \ln _{\kappa }(+\infty )=+\infty }$

${\displaystyle \ln _{\kappa }(1/x)=-\ln _{\kappa }(x)}$

For a real number ${\displaystyle r}$, the κ-logarithm has the property:

${\displaystyle \ln _{\kappa }(x^{r})=r\ln _{r\kappa }(x)}$

For any ${\displaystyle x,y\in \mathbb {R} }$ and ${\displaystyle |\kappa |<1}$, the Kaniadakis sum (or κ-sum) is defined by the following composition law:

${\displaystyle x{\stackrel {\kappa }{\oplus }}y=x{\sqrt {1+\kappa ^{2}y^{2}}}+y{\sqrt {1+\kappa ^{2}x^{2}}}}$,

that can also be written in form:

${\displaystyle x{\stackrel {\kappa }{\oplus }}y={1 \over \kappa }\,\sinh \left({\rm {arcsinh}}\,(\kappa x)\,+\,{\rm {arcsinh}}\,(\kappa y)\,\right)}$,

where the ordinary sum is a particular case in the classical limit ${\displaystyle \kappa \rightarrow 0}$: ${\displaystyle x{\stackrel {0}{\oplus }}y=x+y}$.

The κ-sum, like the ordinary sum, has the following properties:

${\displaystyle {\text{1. associativity:}}\quad (x{\stackrel {\kappa }{\oplus }}y){\stackrel {\kappa }{\oplus }}z=x{\stackrel {\kappa }{\oplus }}(y{\stackrel {\kappa }{\oplus }}z)}$

${\displaystyle {\text{2. neutral element:}}\quad x{\stackrel {\kappa }{\oplus }}0=0{\stackrel {\kappa }{\oplus }}x=x}$

${\displaystyle {\text{3. opposite element:}}\quad x{\stackrel {\kappa }{\oplus }}(-x)=(-x){\stackrel {\kappa }{\oplus }}x=0}$

${\displaystyle {\text{4. commutativity:}}\quad x{\stackrel {\kappa }{\oplus }}y=y{\stackrel {\kappa }{\oplus }}x}$

The κ-difference ${\displaystyle {\stackrel {\kappa }{\ominus }}}$ is given by ${\displaystyle x{\stackrel {\kappa }{\ominus }}y=x{\stackrel {\kappa }{\oplus }}(-y)}$.

The fundamental property ${\displaystyle \exp _{\kappa }(-x)\exp _{\kappa }(x)=1}$ arises as a special case of the more general expression below: ${\displaystyle \exp _{\kappa }(x)\exp _{\kappa }(y)=exp_{\kappa }(x{\stackrel {\kappa }{\oplus }}y)}$

Furthermore, the κ-functions and the κ-sum present the following relationships:

${\displaystyle \ln _{\kappa }(x\,y)=\ln _{\kappa }(x){\stackrel {\kappa }{\oplus }}\ln _{\kappa }(y)}$

For any ${\displaystyle x,y\in \mathbb {R} }$ and ${\displaystyle |\kappa |<1}$, the Kaniadakis product (or κ-product) is defined by the following composition law:

${\displaystyle x{\stackrel {\kappa }{\otimes }}y={1 \over \kappa }\,\sinh \left(\,{1 \over \kappa }\,\,{\rm {arcsinh}}\,(\kappa x)\,\,{\rm {arcsinh}}\,(\kappa y)\,\right)}$,

where the ordinary product is a particular case in the classical limit ${\displaystyle \kappa \rightarrow 0}$: ${\displaystyle x{\stackrel {0}{\otimes }}y=x\times y=xy}$.

The κ-product, like the ordinary product, has the following properties:

${\displaystyle {\text{1. associativity:}}\quad (x{\stackrel {\kappa }{\otimes }}y){\stackrel {\kappa }{\otimes }}z=x{\stackrel {\kappa }{\otimes }}(y{\stackrel {\kappa }{\otimes }}z)}$

${\displaystyle {\text{2. neutral element:}}\quad x{\stackrel {\kappa }{\otimes }}I=I{\stackrel {\kappa }{\otimes }}x=x\quad {\text{for}}\quad I=\kappa ^{-1}\sinh \kappa {\stackrel {\kappa }{\oplus }}x=x}$

${\displaystyle {\text{3. inverse element:}}\quad x{\stackrel {\kappa }{\otimes }}{\overline {x}}={\overline {x}}{\stackrel {\kappa }{\otimes }}x=I\quad {\text{for}}\quad {\overline {x}}=\kappa ^{-1}\sinh(\kappa ^{2}/{\rm {arcsinh}}\,(\kappa x))}$

${\displaystyle {\text{4. commutativity:}}\quad x{\stackrel {\kappa }{\otimes }}y=y{\stackrel {\kappa }{\otimes }}x}$

The κ-division ${\displaystyle {\stackrel {\kappa }{\oslash }}}$ is given by ${\displaystyle x{\stackrel {\kappa }{\oslash }}y=x{\stackrel {\kappa }{\otimes }}{\overline {y}}}$.

The κ-sum ${\displaystyle {\stackrel {\kappa }{\oplus }}}$ and the κ-product ${\displaystyle {\stackrel {\kappa }{\otimes }}}$ obey the distributive law: ${\displaystyle z{\stackrel {\kappa }{\otimes }}(x{\stackrel {\kappa }{\oplus }}y)=(z{\stackrel {\kappa }{\otimes }}x){\stackrel {\kappa }{\oplus }}(z{\stackrel {\kappa }{\otimes }}y)}$.

The fundamental property ${\displaystyle \ln _{\kappa }(1/x)=-\ln _{\kappa }(x)}$ arises as a special case of the more general expression below:

${\displaystyle \ln _{\kappa }(x\,y)=\ln _{\kappa }(x){\stackrel {\kappa }{\oplus }}\ln _{\kappa }(y)}$

Furthermore, the κ-functions and the κ-product present the following relationships:

${\displaystyle \exp _{\kappa }(x){\stackrel {\kappa }{\otimes }}\exp _{\kappa }(y)=\exp _{\kappa }(x\,+\,y)}$

${\displaystyle \ln _{\kappa }(x\,{\stackrel {\kappa }{\otimes }}\,y)=\ln _{\kappa }(x)+\ln _{\kappa }(y)}$

The Kaniadakis differential (or κ-differential) of ${\displaystyle x}$ is defined by:

${\displaystyle \mathrm {d} _{\kappa }x={\frac {\mathrm {d} \,x}{\displaystyle {\sqrt {1+\kappa ^{2}\,x^{2}}}}}}$.

So, the κ-derivative of a function ${\displaystyle f(x)}$ is related to the Leibniz derivative through:

${\displaystyle {\frac {\mathrm {d} f(x)}{\mathrm {d} _{\kappa }x}}=\gamma _{\kappa }(x){\frac {\mathrm {d} f(x)}{\mathrm {d} x}}}$,

where ${\displaystyle \gamma _{\kappa }(x)={\sqrt {1+\kappa ^{2}x^{2}}}}$ is the Lorentz factor. The ordinary derivative ${\displaystyle {\frac {\mathrm {d} f(x)}{\mathrm {d} x}}}$ is a particular case of κ-derivative ${\displaystyle {\frac {\mathrm {d} f(x)}{\mathrm {d} _{\kappa }x}}}$ in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

The Kaniadakis integral (or κ-integral) is the inverse operator of the κ-derivative defined through

${\displaystyle \int \mathrm {d} _{\kappa }x\,\,f(x)=\int {\frac {\mathrm {d} \,x}{\sqrt {1+\kappa ^{2}\,x^{2}}}}\,\,f(x)}$,

which recovers the ordinary integral in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

The Kaniadakis cyclic trigonometry (or κ-cyclic trigonometry) is based on the κ-cyclic sine (or κ-sine) and κ-cyclic cosine (or κ-cosine) functions defined by:

${\displaystyle \sin _{\kappa }(x)={\frac {\exp _{\kappa }(ix)-\exp _{\kappa }(-ix)}{2i}}}$,

${\displaystyle \cos _{\kappa }(x)={\frac {\exp _{\kappa }(ix)+\exp _{\kappa }(-ix)}{2}}}$,

where the κ-generalized Euler formula is

${\displaystyle \exp _{\kappa }(\pm ix)=\cos _{\kappa }(x)\pm i\sin _{\kappa }(x)}$.:

The κ-cyclic trigonometry preserves fundamental expressions of the ordinary cyclic trigonometry, which is a special case in the limit κ → 0, such as:

${\displaystyle \cos _{\kappa }^{2}(x)+\sin _{\kappa }^{2}(x)=1}$

${\displaystyle \sin _{\kappa }(x{\stackrel {\kappa }{\oplus }}y)=\sin _{\kappa }(x)\cos _{\kappa }(y)+\cos _{\kappa }(x)\sin _{\kappa }(y)}$.

The κ-cyclic tangent and κ-cyclic cotangent functions are given by:

${\displaystyle \tan _{\kappa }(x)={\frac {\sin _{\kappa }(x)}{\cos _{\kappa }(x)}}}$

${\displaystyle \cot _{\kappa }(x)={\frac {\cos _{\kappa }(x)}{\sin _{\kappa }(x)}}}$.

The κ-cyclic trigonometric functions become the ordinary trigonometric function in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

κ-Inverse cyclic function

The Kaniadakis inverse cyclic functions (or κ-inverse cyclic functions) are associated to the κ-logarithm:

${\displaystyle {\rm {arcsin}}_{\kappa }(x)=-i\ln _{\kappa }\left({\sqrt {1-x^{2}}}+ix\right)}$,

${\displaystyle {\rm {arccos}}_{\kappa }(x)=-i\ln _{\kappa }\left({\sqrt {x^{2}-1}}+x\right)}$,

${\displaystyle {\rm {arctan}}_{\kappa }(x)=i\ln _{\kappa }\left({\sqrt {\frac {1-ix}{1+ix}}}\right)}$,

${\displaystyle {\rm {arccot}}_{\kappa }(x)=i\ln _{\kappa }\left({\sqrt {\frac {ix+1}{ix-1}}}\right)}$.

The Kaniadakis hyperbolic trigonometry (or κ-hyperbolic trigonometry) is based on the κ-hyperbolic sine and κ-hyperbolic cosine given by:

${\displaystyle \sinh _{\kappa }(x)={\frac {\exp _{\kappa }(x)-\exp _{\kappa }(-x)}{2}}}$,

${\displaystyle \cosh _{\kappa }(x)={\frac {\exp _{\kappa }(x)+\exp _{\kappa }(-x)}{2}}}$,

where the κ-Euler formula is

${\displaystyle \exp _{\kappa }(\pm x)=\cosh _{\kappa }(x)\pm \sinh _{\kappa }(x)}$.

The κ-hyperbolic tangent and κ-hyperbolic cotangent functions are given by:

${\displaystyle \tanh _{\kappa }(x)={\frac {\sinh _{\kappa }(x)}{\cosh _{\kappa }(x)}}}$

${\displaystyle \coth _{\kappa }(x)={\frac {\cosh _{\kappa }(x)}{\sinh _{\kappa }(x)}}}$.

The κ-hyperbolic trigonometric functions become the ordinary hyperbolic trigonometric functions in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

From the κ-Euler formula and the property ${\displaystyle \exp _{\kappa }(-x)\exp _{\kappa }(x)=1}$ the fundamental expression of κ-hyperbolic trigonometry is given as follows:

${\displaystyle \cosh _{\kappa }^{2}(x)-\sinh _{\kappa }^{2}(x)=1}$

κ-Inverse hyperbolic function

The Kaniadakis inverse hyperbolic functions (or κ-inverse hyperbolic functions) are associated to the κ-logarithm:

${\displaystyle {\rm {arcsinh}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {1+x^{2}}}+x\right)}$,

${\displaystyle {\rm {arccosh}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {x^{2}-1}}+x\right)}$,

${\displaystyle {\rm {arctanh}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {\frac {1+x}{1-x}}}\right)}$,

${\displaystyle {\rm {arccoth}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {\frac {1-x}{1+x}}}\right)}$,

in which are valid the following relations:

${\displaystyle {\rm {arcsinh}}_{\kappa }(x)={\rm {sign}}(x){\rm {arccosh}}_{\kappa }\left({\sqrt {1+x^{2}}}\right)}$,

${\displaystyle {\rm {arcsinh}}_{\kappa }(x)={\rm {arctanh}}_{\kappa }\left({\frac {x}{\sqrt {1+x^{2}}}}\right)}$,

${\displaystyle {\rm {arcsinh}}_{\kappa }(x)={\rm {arccoth}}_{\kappa }\left({\frac {\sqrt {1+x^{2}}}{x}}\right)}$.

The κ-cyclic and κ-hyperbolic trigonometric functions are connected by the following relationships:

${\displaystyle {\rm {sin}}_{\kappa }(x)=-i{\rm {sinh}}_{\kappa }(ix)}$,

${\displaystyle {\rm {cos}}_{\kappa }(x)={\rm {cosh}}_{\kappa }(ix)}$,

${\displaystyle {\rm {tan}}_{\kappa }(x)=-i{\rm {tanh}}_{\kappa }(ix)}$,

${\displaystyle {\rm {cot}}_{\kappa }(x)=i{\rm {coth}}_{\kappa }(ix)}$,

${\displaystyle {\rm {arcsin}}_{\kappa }(x)=-i\,{\rm {arcsinh}}_{\kappa }(ix)}$,

${\displaystyle {\rm {arccos}}_{\kappa }(x)\neq -i\,{\rm {arccosh}}_{\kappa }(ix)}$,

${\displaystyle {\rm {arctan}}_{\kappa }(x)=-i\,{\rm {arctanh}}_{\kappa }(ix)}$,

${\displaystyle {\rm {arccot}}_{\kappa }(x)=i\,{\rm {arccoth}}_{\kappa }(ix)}$.

The Kaniadakis statistics is based on the Kaniadakis κ-entropy, which is defined through:

${\displaystyle S_{\kappa }{\big (}p{\big )}=-\sum _{i}p_{i}\ln _{\kappa }{\big (}p_{i}{\big )}=\sum _{i}p_{i}\ln _{\kappa }{\bigg (}{\frac {1}{p_{i}}}{\bigg )}}$

where ${\displaystyle p=\{p_{i}=p(x_{i});x\in \mathbb {R} ;i=1,2,...,N;\sum _{i}p_{i}=1\}}$ is a probability distribution function defined for a random variable ${\displaystyle X}$, and ${\displaystyle 0\leq |\kappa |<1}$ is the entropic index.

The Kaniadakis κ-entropy is thermodynamically and Lesche stable^[22][23] and obeys the Shannon-Khinchin axioms of continuity, maximality, generalized additivity and expandability.

A Kaniadakis distribution (or κ-distribution) is a probability distribution derived from the maximization of Kaniadakis entropy under appropriate constraints. In this regard, several probability distributions emerge for analyzing a wide variety of phenomenology associated with experimental power-law tailed statistical distributions.

The Kaniadakis Laplace transform (or κ-Laplace transform) is a κ-deformed integral transform of the ordinary Laplace transform. The κ-Laplace transform converts a function ${\displaystyle f}$ of a real variable ${\displaystyle t}$ to a new function ${\displaystyle F_{\kappa }(s)}$ in the complex frequency domain, represented by the complex variable ${\displaystyle s}$. This κ-integral transform is defined as:^[24]

${\displaystyle F_{\kappa }(s)={\cal {L}}_{\kappa }\{f(t)\}(s)=\int _{\,0}^{\infty }\!f(t)\,[\exp _{\kappa }(-t)]^{s}\,dt}$

The inverse κ-Laplace transform is given by:

${\displaystyle f(t)={\cal {L}}_{\kappa }^{-1}\{F_{\kappa }(s)\}(t)={{\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }\!F_{\kappa }(s)\,{\frac {[\exp _{\kappa }(t)]^{s}}{\sqrt {1+\kappa ^{2}t^{2}}}}\,ds}}$

The ordinary Laplace transform and its inverse transform are recovered as ${\displaystyle \kappa \rightarrow 0}$.

Properties

Let two functions ${\displaystyle f(t)={\cal {L}}_{\kappa }^{-1}\{F_{\kappa }(s)\}(t)}$ and ${\displaystyle g(t)={\cal {L}}_{\kappa }^{-1}\{G_{\kappa }(s)\}(t)}$, and their respective κ-Laplace transforms ${\displaystyle F_{\kappa }(s)}$ and ${\displaystyle G_{\kappa }(s)}$, the following table presents the main properties of κ-Laplace transform:^[24]

Properties of the κ-Laplace transform

Property	${\displaystyle f(t)}$	${\displaystyle F_{\kappa }(s)}$
Linearity	${\displaystyle a\,f(t)+b\,g(t)}$	${\displaystyle a\,F_{\kappa }(s)+b\,G_{\kappa }(s)}$
Time scaling	${\displaystyle f(at)}$	${\displaystyle {\frac {1}{a}}\,F_{\kappa /a}({\frac {s}{a}})}$
Frequency shifting	${\displaystyle f(t)\,[\exp _{\kappa }(-t)]^{a}}$	${\displaystyle F_{\kappa }(s-a)}$
Derivative	${\displaystyle {\frac {d\,f(t)}{dt}}}$	${\displaystyle s\,{\cal {L}}_{\kappa }\left\{{\frac {f(t)}{\sqrt {1+\kappa ^{2}t^{2}}}}\right\}(s)-f(0)}$
Derivative	${\displaystyle {\frac {d}{dt}}\,{\sqrt {1+\kappa ^{2}t^{2}}}\,f(t)}$	${\displaystyle s\,F_{\kappa }(s)-f(0)}$
Time-domain integration	${\displaystyle {\frac {1}{\sqrt {1+\kappa ^{2}t^{2}}}}\,\int _{0}^{t}f(w)dw}$	${\displaystyle {\frac {1}{s}}\,F_{\kappa }(s)}$
	${\displaystyle f(t)\,[\ln(\exp _{\kappa }(t))]^{n}}$	${\displaystyle (-1)^{n}{\frac {d^{n}F_{\kappa }(s)}{ds^{n}}}}$
	${\displaystyle f(t)\,[\ln(\exp _{\kappa }(t))]^{-n}}$	${\displaystyle \int _{s}^{+\infty }dw_{n}\int _{w_{n}}^{+\infty }dw_{n-1}...\int _{w_{3}}^{+\infty }dw_{2}\int _{w_{2}}^{+\infty }dw_{1}\,F_{\kappa }(w_{1})}$
Dirac delta-function	${\displaystyle \delta (t-\tau )}$	${\displaystyle [\exp _{\kappa }(-\tau )]^{s}}$
Heaviside unit function	${\displaystyle u(t-\tau )}$	${\displaystyle {\frac {s{\sqrt {1+\kappa ^{2}\tau ^{2}}}+\kappa ^{2}\tau }{s^{2}-\kappa ^{2}}}\,[\exp _{\kappa }(-\tau )]^{s}}$
Power function	${\displaystyle t^{\nu -1}}$	${\displaystyle {\frac {s^{2}}{s^{2}-\kappa ^{2}\nu ^{2}}}\,{\frac {\Gamma _{\frac {\kappa }{s}}(\nu +1)}{\nu \,s^{\nu }}}={\frac {s}{s+|\kappa |\nu }}\,{\frac {\Gamma (\nu )}{|2\kappa |^{\nu }}}\,{\frac {\Gamma \left({\frac {s}{|2\kappa |}}-{\frac {\nu }{2}}\right)}{\Gamma \left({\frac {s}{|2\kappa |}}+{\frac {\nu }{2}}\right)}}}$
Power function	${\displaystyle t^{2m-1},\ \ m\in Z^{+}}$	${\displaystyle {\frac {(2m-1)!}{\prod _{j=1}^{m}\left[s^{2}-(2j)^{2}\kappa ^{2}\right]}}}$
Power function	${\displaystyle t^{2m},\ \ m\in Z^{+}}$	${\displaystyle {\frac {(2m)!\,s}{\prod _{j=1}^{m+1}\left[s^{2}-(2j-1)^{2}\kappa ^{2}\right]}}}$

The κ-Laplace transforms presented in the latter table reduce to the corresponding ordinary Laplace transforms in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

The Kaniadakis Fourier transform (or κ-Fourier transform) is a κ-deformed integral transform of the ordinary Fourier transform, which is consistent with the κ-algebra and the κ-calculus. The κ-Fourier transform is defined as:^[25]

${\displaystyle {\cal {F}}_{\kappa }[f(x)](\omega )={1 \over {\sqrt {2\,\pi }}}\int \limits _{-\infty }\limits ^{+\infty }f(x)\,\exp _{\kappa }(-x\otimes _{\kappa }\omega )^{i}\,d_{\kappa }x}$

which can be rewritten as

${\displaystyle {\cal {F}}_{\kappa }[f(x)](\omega )={1 \over {\sqrt {2\,\pi }}}\int \limits _{-\infty }\limits ^{+\infty }f(x)\,{\exp(-i\,x_{\{\kappa \}}\,\omega _{\{\kappa \}}) \over {\sqrt {1+\kappa ^{2}\,x^{2}}}}\,dx}$

where ${\displaystyle x_{\{\kappa \}}={\frac {1}{\kappa }}\,{\rm {arcsinh}}\,(\kappa \,x)}$ and ${\displaystyle \omega _{\{\kappa \}}={\frac {1}{\kappa }}\,{\rm {arcsinh}}\,(\kappa \,\omega )}$. The κ-Fourier transform imposes an asymptotically log-periodic behavior by deforming the parameters ${\displaystyle x}$ and ${\displaystyle \omega }$ in addition to a damping factor, namely ${\displaystyle {\sqrt {1+\kappa ^{2}\,x^{2}}}}$.

The kernel of the κ-Fourier transform is given by:

${\displaystyle h_{\kappa }(x,\omega )={\frac {\exp(-i\,x_{\{\kappa \}}\,\omega _{\{\kappa \}})}{\sqrt {1+\kappa ^{2}\,x^{2}}}}}$

The inverse κ-Fourier transform is defined as:^[25]

${\displaystyle {\cal {F}}_{\kappa }[{\hat {f}}(\omega )](x)={1 \over {\sqrt {2\,\pi }}}\int \limits _{-\infty }\limits ^{+\infty }{\hat {f}}(\omega )\,\exp _{\kappa }(\omega \otimes _{\kappa }x)^{i}\,d_{\kappa }\omega }$

Let ${\displaystyle u_{\kappa }(x)={\frac {1}{\kappa }}\cosh {\Big (}\kappa \ln(x){\Big )}}$, the following table shows the κ-Fourier transforms of several notable functions:^[25]

κ-Fourier transform of several functions

	${\displaystyle f(x)}$	${\displaystyle {\cal {F}}_{\kappa }[f(x)](\omega )}$
Step function	${\displaystyle \theta (x)}$	${\displaystyle {\sqrt {2\,\pi }}\,\delta (\omega )+{1 \over {\sqrt {2\,\pi }}\,i\,\omega _{\{\kappa \}}}}$
Modulation	${\displaystyle \cos _{\kappa }(a{\stackrel {\kappa }{\oplus }}x)}$	${\displaystyle {\sqrt {\pi \over 2}}\,u_{\kappa }(\exp _{\kappa }(a))\,\left(\delta (\omega +a)+\delta (\omega -a)\right)}$
Causal ${\displaystyle \kappa }$-exponential	${\displaystyle \theta (x)\,\exp _{\kappa }(-a{\stackrel {\kappa }{\otimes }}x)}$	${\displaystyle {1 \over {\sqrt {2\,\pi }}}{1 \over a_{\{\kappa \}}+i\,\omega _{\{\kappa \}}}}$
Symmetric ${\displaystyle \kappa }$-exponential	${\displaystyle \exp _{\kappa }(-a{\stackrel {\kappa }{\otimes }}|x|)}$	${\displaystyle {\sqrt {2 \over \pi }}\,{a_{\{\kappa \}} \over a_{\{\kappa \}}^{2}+\omega _{\{\kappa \}}^{2}}}$
Constant	${\displaystyle 1}$	${\displaystyle {\sqrt {2\,\pi }}\,\delta (\omega )}$
${\displaystyle \kappa }$-Phasor	${\displaystyle \exp _{\kappa }\,(a{\stackrel {\kappa }{\otimes }}x)^{i}}$	${\displaystyle {\sqrt {2\,\pi }}\,u_{\kappa }(\exp _{\kappa }(a))\,\delta (\omega -a)}$
Impuslse	${\displaystyle \delta (x-a)}$	${\displaystyle {1 \over {\sqrt {2\,\pi }}}{\exp _{\kappa }\,(\omega {\stackrel {\kappa }{\otimes }}a)^{i} \over u_{\kappa }\left(\exp _{\kappa }\,(a)\right)}}$
Signum	Sgn${\displaystyle (x)}$	${\displaystyle {\sqrt {2 \over \pi }}\,\,{1 \over i\,\omega _{\{\kappa \}}}}$
Rectangular	${\displaystyle \Pi \left({x \over a}\right)}$	${\displaystyle {\sqrt {2 \over \pi }}\,\,a_{\{\kappa \}}\,{\rm {sinc}}_{\kappa }(\omega {\stackrel {\kappa }{\otimes }}a)}$

The κ-deformed version of the Fourier transform preserves the main properties of the ordinary Fourier transform, as summarized in the following table.

κ-Fourier properties

${\displaystyle f(x)}$	${\displaystyle {\cal {F}}_{\kappa }[f(x)](\omega )}$
Linearity	${\displaystyle {\cal {F}}_{\kappa }[\alpha \,f(x)+\beta \,g(x)](\omega )=\alpha \,{\cal {F}}_{\kappa }[f(x)](\omega )+\beta \,{\cal {F}}_{\kappa }[g(x)](\omega )}$
Scaling	${\displaystyle {\cal {F}}_{\kappa }\left[f(\alpha \,x)\right](\omega )={1 \over \alpha }\,{\cal {F}}_{\kappa ^{\prime }}\left[f(x)\right](\omega ^{\prime })}$ where ${\displaystyle \kappa ^{\prime }=\kappa /\alpha }$ and ${\displaystyle \omega ^{\prime }=(a/\kappa )\,\sinh \left({\rm {arcsinh}}(\kappa \,\omega )/a^{2}\right)}$
${\displaystyle \kappa }$-Scaling	${\displaystyle {\cal {F}}_{\kappa }\left[f(\alpha {\stackrel {\kappa }{\otimes }}x)\right](\omega )={1 \over \alpha _{\{\kappa \}}}\,{\cal {F}}_{\kappa }[f(x)]\left({\frac {1}{\alpha }}{\stackrel {\kappa }{\otimes }}\omega \right)}$
Complex conjugation	${\displaystyle {\cal {F}}_{\kappa }{\big [}f(x){\big ]}^{\ast }(\omega )={\cal {F}}_{\kappa }{\big [}f(x){\big ]}(-\omega )}$
Duality	${\displaystyle {\cal {F}}_{\kappa }{\Big [}{\cal {F}}_{\kappa }{\big [}f(x){\big ]}(\nu ){\Big ]}(\omega )=f(-\omega )}$
Reverse	${\displaystyle {\cal {F}}_{\kappa }\left[f(-x)\right](\omega )={\cal {F}}_{\kappa }[f(x)](-\omega )}$
${\displaystyle \kappa }$-Frequency shift	${\displaystyle {\cal {F}}_{\kappa }\left[\exp _{\kappa }(\omega _{0}{\stackrel {\kappa }{\otimes }}x)^{i}f(x)\right](\omega )={\cal {F}}_{\kappa }[f(x)](\omega {\stackrel {\kappa }{\ominus }}\omega _{0})}$
${\displaystyle \kappa }$-Time shift	${\displaystyle {\cal {F}}_{\kappa }\left[f(x\,{\stackrel {\kappa }{\oplus }}\,x_{0})\right](\omega )=\exp _{\kappa }(\omega \,{\stackrel {\kappa }{\otimes }}\,x_{0})^{i}\,{\cal {F}}_{\kappa }[f(x)](\omega )}$
Transform of ${\displaystyle \kappa }$-derivative	${\displaystyle {\cal {F}}_{\kappa }\left[{\frac {d\,f(x)}{d_{\kappa }x}}\right](\omega )=i\,\omega _{\{\kappa \}}\,{\cal {F}}_{\kappa }[f(x)](\omega )}$
${\displaystyle \kappa }$-Derivative of transform	${\displaystyle {\frac {d}{d_{\kappa }\omega }}\,{\cal {F}}_{\kappa }[f(x)](\omega )=-i\,\omega _{\{\kappa \}}\,{\cal {F}}_{\kappa }\left[x_{\{\kappa \}}\,f(x)\right](\omega )}$
Transform of integral	${\displaystyle {\cal {F}}_{\kappa }\left[\int \limits _{-\infty }\limits ^{x}f(y)\,dy\right](\omega )={1 \over i\,\omega _{\{\kappa \}}}{\cal {F}}_{\kappa }[f(x)](\omega )+2\,\pi \,{\cal {F}}_{\kappa }[f(x)](0)\,\delta (\omega )}$
${\displaystyle \kappa }$-Convolution	${\displaystyle {\cal {F}}_{\kappa }\left[(f\,{\stackrel {\kappa }{\circledast }}\,g)(x)\right](\omega )={\sqrt {2\,\pi }}\,{\cal {F}}_{\kappa }[f(x)](\omega )\,{\cal {F}}_{\kappa }[g(x)](\omega )}$ where ${\displaystyle (f\,{\stackrel {\kappa }{\circledast }}\,g)(x)=\int \limits _{-\infty }\limits ^{+\infty }f(y)\,g(x\,{\stackrel {\kappa }{\ominus }}\,y)\,d_{\kappa }y}$
Modulation	${\displaystyle {\cal {F}}_{\kappa }\left[f(x)\,g(x)\right](\omega )={1 \over {\sqrt {2\,\pi }}}\left({\cal {F}}_{\kappa }\left[f(x)\right]\,{\stackrel {\kappa }{\circledast }}\,{\cal {F}}_{\kappa }\left[g(x)\right]\right)(\omega )}$

The properties of the κ-Fourier transform presented in the latter table reduce to the corresponding ordinary Fourier transforms in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

• Giorgio Kaniadakis
• Kaniadakis distribution
• Kaniadakis κ-Exponential distribution
• Kaniadakis κ-Gaussian distribution
• Kaniadakis κ-Gamma distribution
• Kaniadakis κ-Weibull distribution
• Kaniadakis κ-Logistic distribution
• Kaniadakis κ-Erlang distribution

•  This article incorporates text available under the CC BY 3.0 license.

• Giorgio Kaniadakis Google Scholar page
• Kaniadakis Statistics on arXiv.org