n-tuple Notation

I am reading a paper (to be able to implement the Baum-Welch algorithm in it) and the following notation is defined:

$$[ a_k ]_{k=i}^j ≡ (a_i, a_{i+1}, \ldots , a_j)$$

$$[a(k)]_{k=i}^j ≡ (a(i), a(i+ 1), \ldots , a( j))$$

I (think) the first is shorthand of a n-tuple. I guess the second is something like that, but I don’t understand the difference between the first and the second. Is this common notation something odd, or what? I am pretty rusty on my math, so if I am missing something obvious please don’t hesitate to point that out.

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As the comments state, there is no essential difference between the two. They simply both mean:

“The part of the sequence $a$ starting at index $i$ and ending at $j$.”

As Brian M. Scott points out, there are several justifications for both of these notations for a sequence occurring:

Sometimes you want to emphasize the fact that a is a function of $k$. Sometimes the choice is made on the basis of readability: one version combines badly with other notation in use, resulting in cluttered expressions that are hard to read. Sometimes you just happened to use one instead of the other.