ENH: support pathlib.Path division in StringArray

[jbrockmendel]

• closes BUG: Using / operator with a Path and Series of string dtype fails #61940 (Replace xxxx with the GitHub issue number)
• Tests added and passed if fixing a bug or adding a new feature
• All code checks passed.
• Added type annotations to new arguments/methods/functions.
• Added an entry in the latest doc/source/whatsnew/vX.X.X.rst file if fixing a bug or adding a new feature.

[mroeschke]

Thanks @jbrockmendel

[lumberbot-app]

Owee, I'm MrMeeseeks, Look at me.

There seem to be a conflict, please backport manually. Here are approximate instructions:

1. Checkout backport branch and update it.

git checkout 2.3.x
git pull

2. Cherry pick the first parent branch of the this PR on top of the older branch:

git cherry-pick -x -m1 5b16660a33a2b2457ea0d60297698a6040f7eb5e

3. You will likely have some merge/cherry-pick conflict here, fix them and commit:

git commit -am 'Backport PR #62229: ENH: support pathlib.Path division in StringArray'

4. Push to a named branch:

git push YOURFORK 2.3.x:auto-backport-of-pr-62229-on-2.3.x

5. Create a PR against branch 2.3.x, I would have named this PR:

"Backport PR #62229 on branch 2.3.x (ENH: support pathlib.Path division in StringArray)"

And apply the correct labels and milestones.

Congratulations — you did some good work! Hopefully your backport PR will be tested by the continuous integration and merged soon!

Remember to remove the Still Needs Manual Backport label once the PR gets merged.

If these instructions are inaccurate, feel free to suggest an improvement.

[jorisvandenbossche]

Manual backport -> #62239