What does a dollar sign followed by a square bracket $[...] mean in bash?

In a thread on the GNU bash mailing list, it says that the $[ syntax was an early syntax that was deprecated in favor of $((, since the latter was already used by the Korn shell.

According to this site, the manual for bash 3.2.48 contained a reference to the $[ syntax. So presumably this reference had been removed in 3.2.51.

You can find old bash source here. In particular I downloaded bash-1.14.7.tar.gz. In the documentation/bash.txt you will find:

Arithmetic Expansion

Arithmetic expansion allows the evaluation of an arithmetic expression and the substitution of the result. There are two formats for arithmetic expansion:

     $[expression]

     $((expression))

The references to $[ are gone in doc/bash.html from the bash-doc-2.0.tar.gz download and the NEWS file mentions that:

The $[...] arithmetic expansion syntax is no longer supported, in favor of $((...)).

$((...)) is also the standard syntax for an arithmetic expansion, but may have been added to the standard later than the original Bash implementation.

However, $[...] does still seem to work in Bash 5.0, so it's not completely removed.