What are the parentheses used for in a bash shell script function definition like "f () {}"? Is it different than using the "function" keyword?

The keyword function has been deprecated in favor of function_name() for portability with the POSIX spec

A function is a user-defined name that is used as a simple command to call a compound command with new positional parameters. A function is defined with a "function definition command".

The format of a function definition command is as follows:

fname() compound-command[io-redirect ...]

Note that the { } are not mandatory so if you're not going to use the keyword function (and you shouldn't) then the () are necessary so the parser knows you're defining a function.

Example, this is a legal function definition and invocation:

$ myfunc() for arg; do echo "$arg"; done; myfunc foo barfoobar

The empty parentheses are required in your first example so that bash knows it's a function definition (otherwise it looks like an ordinary command). In the second example, the () is optional because you've used function.

Without function, alias expansion happens at definition time. E.g.:

alias a=b# Gets expanded to "b() { echo c; }" :a() { echo c; }b# => c# Gets expanded to b:a# => c

With function however, alias expansion does not happen at definition time, so the alias "hides" the definition:

alias a=bfunction a { echo c; }b# => command not found# Gets expanded to b:a# => command not foundunalias aa# => c