I'm curious about the importance and applications of the Hausdorff's Formula: $$\aleph_{\alpha+1}^{\aleph_\beta}=\aleph_{\alpha}^{\aleph_\beta}.\aleph_{\alpha+1}$$ What are its implications? Where can I use it?
I was wondering if it's used to prove some important result, or if there is some paper about it.
An answer to your question could be just as generalistic as your question itself really is.
The Hausdorff recursion formula builds the fundament of what Hausdorff himself nominated as confinality and subsequently every other result that deducted versus Aleph exponentaition.
Historically favorable, would suggest you start the journey with his own symphony "H.: Gesammelte Werke. Band II: Grundzüge der Mengenlehre. Springer-Verlag, Berlin, Heidelberg". There should be perhaps an English version.
To your question in comment, tha answer is yes and the list is long. The latest I am aware of, is the recursive formula of Silver in: J. Silver, "On the singular cardinals problem" R. James (ed.) , Proc. Internat. Congress Mathematicians (Vancouver, 1974) , 1 , Canad. Math. Congress (1975) pp. 265–268
The simplest deduction should be the well known Bernshtein formula.