I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for context) by hypothesizing and using the first few terms of the Baker formula to verify. In order to prove my result rigorously, I think I need the nth term of the Baker formula. Is there an expression for that - I could not find it through online research. ? It would also help if I could find a proof of the BCH formula which is based on recursion i.e. if I could see how the nth term relates recursively to the (n-1)th term.
The Dynkin formula is somewhat cumbersome. Maybe a better choice is Goldberg's version http://projecteuclid.org/euclid.dmj/1077466673 In the commutator form Goldberg's result is reformulated in http://www.ams.org/journals/proc/1982-086-01/S0002-9939-1982-0663855-0/ (Cyclic relations and the Goldberg coefficients in the Campbell-Baker-Hausdorff formula, by Robert C. Thompson).
By the way an interesting early history of the Baker-Campbell-Hausdorff-Dynkin formula can be found in http://link.springer.com/article/10.1007%2Fs00407-012-0095-8 (The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin, by R. Achilles and A. Bonfiglioli).
There is a formula due to Dynkin (1947). It can for instance be found as Theorem 2.2 in these lecture notes. You may also google Baker-Campbell-Hausdorff-Dynkin formula.
A completely explicit formula is obtained in a nice way in Bourbaki, Lie Groups and Lie Algebras, ch. 2, §6.