(n+1)! = n! * (n+1)

Hi,

how can such definitons like factorial or other products be calculated  
in a way,
that allows transformations like that, mentioned in the subject?

(I would like to have both directions.)

How can other formulas be handled in a (meta-?)symbolic way,
for example iterations on functions and derivatives handling with index-values
from 1 to n, with "..." for the parts between 1 and n in formulas.

What is needed, is things like Sum-formulas, for example Taylor of a  
function f,
and the sum not with \sum symbol but with f^{(1)}, f^{(2)}, ..., f^{(n)},
and if I would use derivative of that taylor term, then the  
derivation-index is changed accordingly.

Is such things possible?

Can the factorial-example be done with built-in functions?
or with recursive definitions of user-specific functions?
Any way to handle such things with axiom?

It would make some proofs (induction), which habe a lot of factorial-terms
or the above mentioned sum terms, easier to devleop adn write down  
intuitively.


TIA,
   pan