{"title":"Explicit Expressions for Iterates of Power Series","authors":"Beauduin Kei","doi":"arxiv-2409.09809","DOIUrl":null,"url":null,"abstract":"In this paper, we present five different formulas for both discrete and\nfractional iterations of an invertible power series $f$ utilizing a novel and\nunifying approach from umbral calculus. Established formulas are extended, and\ntheir proofs simplified, while new formulas are introduced. In particular,\nthrough the use of $q$-calculus identities, we eliminate the requirement for\n$f'(0)$ to equal $1$ and, consequently, the corresponding new expressions for\nthe iterative logarithm are derived.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09809","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we present five different formulas for both discrete and
fractional iterations of an invertible power series $f$ utilizing a novel and
unifying approach from umbral calculus. Established formulas are extended, and
their proofs simplified, while new formulas are introduced. In particular,
through the use of $q$-calculus identities, we eliminate the requirement for
$f'(0)$ to equal $1$ and, consequently, the corresponding new expressions for
the iterative logarithm are derived.
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