Gaussian bell

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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 $$\mathbf{E}(t)=e_{p}(t,0).$$

Properties[edit]

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

References[edit]