Is it appropriate to request letters of recommendation from professors a second time? I received three letters of recommendation for applications to this spring semester, but I chose to decline admission for personal/financial reasons. I am now applying for fall admission, and I would like to ask the same professors for letters of recommendation again.

Is it appropriate to request additional letters of recommendations from professors to new universities for the second time? With no other prospects, I need their recommendations in order to be accepted. Should that desperation be included or excluded in the request?
Yes.

Writing recommendation letters is part of our job as professors.  It is perfectly ethical to request that we do our job.  

Conveying desperation in your request is neither necessary nor productive.  You have nothing to be ashamed of or to apologize for.

On the other hand, anyone can (and should!) turn down your request in good conscience if they feel that they cannot write you a strong recommendation letter.  If someone says that they can't write you a strong letter, believe them and ask someone else.  You really don't want a weak letter in your file.