Consider the Baker–Campbell–Hausdorff formula ([Wikipedia page](https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula)):
$$Z(X,Y) := X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + \dotsb$$


Many sources, including the Wikipedia page, have an explicit expression for the terms, so the question I'm asking should be answerable just from that expression.


For a prime $p$ and a natural number $n$, denote by $f(p,n)$ the largest $k$ such that, if we truncate the formula to terms that involve products of length at most $n$, then one or more of the denominators is divisible by $p^k$. Note that this truncation is valid when we are working in a situation of nilpotency class $n$.


It's pretty easy to see that $f(p,n) = 0$ for $n < p$, and is nonzero for $n \ge p$.


My question: Is there a direct explicit expression for $f(p,n)$ (or a sandwiching of it between two fairly close expressions)? For instance, inspection of the first few terms suggests that $f(2,n) = n - 1$, but I'm not sure how to derive this from the general expression.


Analogue: In the power series for the exponential $e^x$, the analogue to $f(p,n)$ is the sum $[n/p] + [n/p^2] + [n/p^3] + \dotsb$ where $[\,\cdot\,]$ denotes the greatest integer function.


UPDATE: [Chapter 3](https://doi.org/10.1007/BFb0094089 "p-adically simple groups (p~−groups)") of the Springer Lecture Notes in Mathematics [vol. 1674](https://doi.org/10.1007/BFb0094086 "Linear Pro-p-Groups of Finite Width. Berlin, Springer (1997). zbMATH review at https://zbmath.org/?q=an:0901.20013") by Klaas, Leedham-Green, and Plesken contains some estimates. However: (i) I'm not sure all the numerical calculations there are correct, since they don't agree with others I have seen, (ii) the authors aren't concerned about the precise growth of $f(p,n)$ — they only care that it grows slowly enough that the series converges under certain conditions.



Baker–Campbell–Hausdorff formula: prime divisors of denominators
If the homogeneous component $Z\_n(X,Y)$ of $Z(X,Y)$ of degree $n$ is represented in the Lyndon basis $\mathcal{L}\_n$, or in any basis $\mathcal{B}\_n$ whose transformation
matrix $T\_{\mathcal{L\_n}\to\mathcal{B\_n}}$ has determinant $\pm 1$, then an explicit formula for the exponent $f(p,n)$ of the highest power of $p$ that divides the least common multiple of the denominators of the coefficients of $Z\_n(X,Y)$ is given by
$$
f(p,n)=\frac{n-s\_p(n)}{p-1}+\lfloor\log\_p(s\_p(n))\rfloor.
$$
Here, $s\_p(n)=\alpha\_0+\ldots+\alpha\_r$ denotes the sum of the digits in the $p$-adic expansion $n=\alpha\_0+\alpha\_1p+\ldots+\alpha\_r p^r$, $0\leq\alpha\_i<p$, and $\log\_p$ is the logarithm to base $p$.


This follows in a straightforward way from the analogous statement for the coefficients of $Z\_n(X,Y)$ represented as a polynomial in the non-commuting variables $X$, $Y$, which was proved in



> 
> Harald Hofstätter, Smallest common denominators for the homogeneous components of the Baker-Campbell-Hausdorff series, [arXiv:2010.03818](https://arxiv.org/abs/2012.03818).
> 
> 
> 


Bases $\mathcal{B}\_n$ with transformation matrix $T\_{\mathcal{L\_n}\to\mathcal{B\_n}}$ with determinant $\pm 1$ include all Hall bases, and also the [rightnormed basis of E.S.Chibrikov](https://www.sciencedirect.com/science/article/pii/S0021869306002572).


