New representations of pi and Dirac delta using the nonextensive-statistical-mechanics q-exponential function

TL;DR

This paper introduces new representations of the Dirac delta and pi using the nonextensive-statistical-mechanics q-exponential function, revealing properties like normalizability for certain q-values.

Contribution

It generalizes the plane wave representation of the Dirac delta and introduces novel pi representations using the q-exponential, expanding the mathematical tools in nonextensive statistics.

Findings

q-exponential representation of delta function

Normalizable q-plane waves for 1<q<3

New families of pi representations

Abstract

We present a generalization of the representation in plane waves of Dirac delta, $\delta(x)=(1/2\pi)\int_{-\infty}^\infty e^{-ikx}\,dk$, namely $\delta(x)=(2-q)/(2\pi)\int_{-\infty}^\infty e_q^{-ikx}\,dk$, using the nonextensive-statistical-mechanics $q$-exponential function, $e_q^{ix}\equiv[1+(1-q)ix]^{1/(1-q)}$ with $e_1^{ix}\equiv e^{ix}$, being $x$ any real number, for real values of $q$ within the interval $[1,2[$. Concomitantly with the development of these new representations of Dirac delta, we also present two new families of representations of the transcendental number $\pi$. Incidentally, we remark that the $q$-plane wave form which emerges, namely $e_q^{ikx}$, is normalizable for $1<q<3$, in contrast with the standard one, $e^{ikx}$, which is not.