Taylor expansions of $R$-transforms, application to supports and moments Florent Benaych-Georges 60E1046L5460E07R-transformfree cumulantsfree infinitely divisible distributions We prove that a probability measure on the real line has a moment of order (even integer), if and only if its $R$-transform admits a Taylor expansion with $p$ terms. We also prove a weaker version of this result when $p$ is odd. We then apply this to prove that a probability measure whose $R$-transform extends analytically to a ball with center zero is compactly supported, and that a free infinitely divisible distribution has a moment of even order $p$, if and only if its L\'evy measure does so. We also prove a weaker version of the last result when $p$ is odd. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2691 10.1512/iumj.2006.55.2691 en Indiana Univ. Math. J. 55 (2006) 465 - 482 state-of-the-art mathematics http://iumj.org/access/