In the original tSNE paper (van der Maaten and Hinton 2008, Visualizing Data using t-SNE) the similarity probabilities for stochastic neighbor embedding (SNE) are defined in Section 2 as
$$p_{j|i} = \frac{\exp(-||x_{i} - x_{j}||/2\sigma_{i}^{2})}{\sum_{k \neq i}{\exp(-||x_{i} - x_{k}||/2\sigma_{i}^{2})}}$$
and the probabilities for t-distributed stochastic neighbor embedding (t-SNE) are defined in Section 3 as
$$p_{ij} = \frac{\exp(-||x_{i} - x_{j}||/2\sigma^{2})}{\sum_{k \neq l}{\exp(-||x_{k} - x_{l}||/2\sigma^{2})}}.$$
According to my prior understanding original, SNE and tSNE differ only in the formula for $q_{ij}$: SNE uses Gaussian for $q_{ij}$ and tSNE uses Student's t-distribution. But the above two formulas are different as well; why?
My questions are about second formula: from where $k$ and $l$ iterators are came from? And that is $\sigma$ is it $\sigma_{i}$ or not? Iterators $k$ and $l$ for the $q_{ij}$ are also confusing me.