It's a known fact that every LTL formula can be expressed by a Büchi $\omega$-automaton. But, apparently, Büchi automata are a more powerful, expressive model. I've heard somewhere that Büchi automata are equivalent to linear-time $\mu$-calculus (that is, $\mu$-calculus with usual fixpoints and only one temporal operator: $\mathbf{X}$).
Is there an algorithm (constructive proof) of this equality?
The constructive equivalence of linear-time fixed point formulae (the logic is called $\nu$TL by some) and Buechi automata is given in a paper by Mads Dam from 1992.
Fixed Points of Buchi Automata, FST&TCS 1992.
See page 4 for the construction of a $\nu$TL formula from a Buechi automaton. The construction of a Buechi automaton from a $\nu$TL formula is more complicated and takes the rest of the paper.
The rest of this reply is a brief argument that this result existed in the literature in far less direct form. Pierre Wolper showed that there were omega-regular properties that were not LTL-definable and gave an extension of LTL (called ETL) that could express omega-regular properties.
Temporal Logic can be more expressive, Pierre Wolper, Information and Computation, 1983.
It is also known that one can translate ETL formulae into $\nu$TL formulae, so by combining these results you can read off a translation of Buechi automata into $\nu$TL. In the other direction, it follows from the work of Buechi that S1S (the second order theory of one successor) formulae can be compiled into Buechi automata and by translating $\nu$TL formulae into S1S, we obtain a translation of $\nu$TL to Buechi automata. If you want a more in-depth introduction to these topics, I suggest Mads Dam's lecture notes, or the work of Roope Kaivola (sadly not as widely known as much related work).
Temporal Logics, Automata, and Classical Theories - An Introduction, Mads Dam, ESSLLI 1994.
Using Automata to Characterise Fixed Point Temporal Logics, Roope Kaivola
IIRC, there was a similar problem in Moshe Vardi's talk at the Fields Institute (it wasn't about $\mu$-calculus though).
You may want to check the slides or check Vardi's papers. There is definitely an algorithm but IIRC the negations cause a huge increase in the time required to perform the translation.
