Formulas for functions of ordered operators

In an algebra with a lattice of functions of ordered elements (e.g., in an algebra of operators), we investigate the expansions of functions of the type $f(A+B)$ and $\varphi(\stackrel1A,\stackrel2B)$ in powers of the commutators $A$, $B$. In particular, we obtain all the terms of the expansion
$$ f(A+B)=f(\stackrel1A+\stackrel2B)+\frac12\stackrel2{\overline{[A,B]}}f^{(2)}(\stackrel1A+\stackrel3B)+\dots $$
A diagram method for a similar type of calculation is developed. Our discussion is based on Maslov's technique of ordered operators.