2023/06/27 by Dariusz Bugajewski, Bugajewski, Dariusz, Dawid Bugajewski +5
Computer Science · #13F25 #13J05 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation #Primary: 05A10 #Secondary: 40A30
paper · pdf · doi:10.48550/arxiv.2306.15750
openalex publication_date 2023/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The famous J.C.P. Miller formula provides a recurrence algorithm for the composition Ba ∘ f, where Ba 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 Ba ∘ f. We also provide the general J.C.P. Miller recurrence algorithm for computing the coefficients of that composition, if Ba∘ 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.