# Gaussian bell

Let $\mathbb{T}$ be a time scale with $0 \in \mathbb{T}$. Let $p \colon \mathbb{T} \rightarrow \mathbb{R}$ be regressive and defined by $$p(t)=\ominus(t \odot 1).$$ The Gaussian bell $\mathbf{E} \colon \mathbb{T} \rightarrow \mathbb{R}$ is defined<ref name=gaussbell /> to be the exponential function $$\mathbf{E}(t)=e_{p}(t,0).$$

 $\mathbb{T}$ $\mathbf{E}(t)$ $\mathbb{R}$ $e^{-\frac{t^2}{2}}$ $\mathbb{Z}$ $2^{\frac{-t(t-1)}{2}}$ $h\mathbb{Z}$ $\left[(1+h)^{\frac{1}{h}} \right]^{\frac{-t(t-h)}{2}}$ $\mathbb{Z}^2$ $\overline{q^{\mathbb{Z}}}, q > 1$ $\overline{q^{\mathbb{Z}}}, q < 1$ $\displaystyle\prod_{k=\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$ $\mathbb{H}$ $\displaystyle\prod_{k=1}^n \left( \dfrac{k}{k+1} \right)^{H_{k-1}}$

## References

<references> <ref name=gaussbell>Erbe, L.; Peterson, A.;Simon, M. Square integrability of Gaussian bells on time scales. Comput. Math. Appl. 49 (2005), no. 5-6, 871--883. </ref> </references>