On the recursive and explicit form of the general J.C.P. Miller formula with applications
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916912377626624 |
|---|---|
| author | Bugajewski, Dariusz Bugajewski, Dawid Gan, Xiao-Xiong Maćkowiak, Piotr |
| author_facet | Bugajewski, Dariusz Bugajewski, Dawid Gan, Xiao-Xiong Maćkowiak, Piotr |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15750 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the recursive and explicit form of the general J.C.P. Miller formula with applications Bugajewski, Dariusz Bugajewski, Dawid Gan, Xiao-Xiong Maćkowiak, Piotr Commutative Algebra Combinatorics Primary: 05A10, 13F25, 13J05, Secondary: 40A30 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. |
| title | On the recursive and explicit form of the general J.C.P. Miller formula with applications |
| topic | Commutative Algebra Combinatorics Primary: 05A10, 13F25, 13J05, Secondary: 40A30 |
| url | https://arxiv.org/abs/2306.15750 |