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

We present a generalization of the representation in plane waves of Dirac delta, δ ( x ) = ( 1 / 2 π ) ∫ − ∞ ∞ e − i k x d k , namely, δ ( x ) = [ ( 2 − q ) / 2 π ] ∫ − ∞ ∞ e q − i k x d k , using the non-extensive-statistical-mechanics q -exponential function, e q i x ≡ [ 1 + ( 1 − q ) i x ] 1 / ( 1 − q ) with e 1 i x ≡ e i x , x being 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 π . Incidentally, we remark that the q -plane wave form which emerges, namely, e q i k x , is normalizable for 1 < q < 3 , in contrast to the standard one, e i k x , which is not.