  The famous J.C.P. Miller formula provides a recurrence algorithm for the
composition $B_a \circ f$, where $B_a$ is the formal binomial series and $f$ is
a formal power series, however it requires that $f$ has to be a nonunit. In
this paper we provide the general J.C.P. Miller formula which eliminates the
requirement of nonunitness of $f$ and, instead, we establish a necessary and
sufficient condition for the existence of the composition $B_a \circ f$. We
also provide the general J.C.P. Miller recurrence algorithm for computing the
coefficients of that composition, if $ B_a\circ f$ is well defined, obviously.
Our generalizations cover both the case in which $f$ is a one--variable formal
power series and the case in which $f$ is a multivariable formal power series.
In the central part of this article we state, using some combinatorial
techniques, the explicit form of the general J.C.P. Miller formula for
one-variable case. As applications of these results we provide an explicit
formula for the inverses of polynomials and formal power series for which the
inverses exist, obviously. We also use our results to investigation of
approximate solution to a differential equation which cannot be solved in an
explicit way.
