by Wei Xi Fan
R: What is
R: What is the quantum analogue of
L: I have no clue.
R: Let’s try This is the sequence
L: OK. . I can’t get anywhere from here that would make it look remotely close to .
R: What do we do?
L: We call upon the fire of inspiration itself, Ignis.
IGNIS: The truth ye seek is but a cascading staircase, but one that ends.
L: What do you think that means?
R: Instead of trying , let’s try
there are terms in total just like , but here, each term is one less than the previous.
L: Like a factorial that suddenly ends. All right, so
Oh! The factors line up. The subtraction yields
This is exactly what we are looking for.
R: Great! So we have the identity
in the real calculus.
L: OK. What about when or when ?
R: Well… what do we do to get from to
L: Well, we divide by its last term, which is , to get to , because the only difference between them is that is missing the term.
R: What do we get when we go from to
L: Well, , and so we divide this by to get to , so I guess we should define
R: What about
L: Well, we would divide this by so I guess it is
R: So what’s the general formula?
Are you sure the identity (*) still holds for these new definitions?
R: For these situations, we should call upon the minion of non-illuminating bashing, Grunt.
GRUNT: ‘Tis done.
R: There, this fact has been verified.
L: I shall call these Pochhammer symbols.
L: Or we can call them falling factorials. Or falling powers.