  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.
