On the recursive and explicit form of the general J.C.P. Miller formula with applications

TL;DR

This paper generalizes the J.C.P. Miller formula to include cases where the formal power series is a unit, providing recursive and explicit formulas for composition and applications to polynomial inverses and differential equations.

Contribution

It introduces a comprehensive version of the J.C.P. Miller formula that removes previous restrictions, applicable to multivariable series and includes explicit formulas and algorithms.

Findings

Derived a necessary and sufficient condition for the existence of the composition.

Provided explicit formulas for polynomial and formal power series inverses.

Developed applications to approximate solutions of differential equations.

Abstract

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.