  We discuss certain aspects of the formal calculus used to describe vertex
algebras. In the standard literature on formal calculus, the expression
$(x+y)^{n}$, where $n$ is not necessarily a nonnegative integer, is defined as
the formal Taylor series given by the binomial series in nonnegative powers of
the second-listed variable (namely, $y$). We present a viewpoint that for some
purposes of generalization of the formal calculus including and beyond
"logarithmic formal calculus", it seems useful, using the formal Taylor theorem
as a guide, to instead take as the definition of $(x+y)^{n}$ the formal series
which is the result of acting on $x^{n}$ by a formal translation operator, a
certain exponentiated derivation. These differing approaches are equivalent,
and in the standard generality of formal calculus or logarithmic formal
calculus there is no reason to prefer one approach over the other. However,
using this second point of view, we may more easily, and in fact do, consider
extensions in two directions, sometimes in conjunction. The first extension is
to replace $x^{n}$ by more general objects such as the formal variable $\log
x$, which appears in the logarithmic formal calculus, and also, more
interestingly, by iterated-logarithm expressions. The second extension is to
replace the formal translation operator by a more general formal change of
variable operator. In addition, we note some of the combinatorics underlying
the formal calculus which we treat, and we end by briefly mentioning a
connection to Fa\`a di Bruno's classical formula for the higher derivatives of
a composite function and the classical umbral calculus. Many of these results
are extracted from more extensive papers \cite{R1} and \cite{R2}, to appear.
