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Fractional power series and the method of dominant balances

Chapman, C. J. and Wynn, H. P. ORCID: 0000-0002-6448-1080 (2021) Fractional power series and the method of dominant balances. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477 (2246). ISSN 1364-5021

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Identification Number: 10.1098/rspa.2020.0646


This paper derives an explicit formula for a type of fractional power series, known as a Puiseux series, arising in a wide class of applied problems in the physical sciences and engineering. Detailed consideration is given to the gaps which occur in these series (lacunae); they are shown to be determined by a number-theoretic argument involving the greatest common divisor of a set of exponents appearing in the Newton polytope of the problem, and by two number-theoretic objects, called here Sylvester sets, which are complements of Frobenius sets. A key tool is Faà di Bruno's formula for high derivatives, as implemented by Bell polynomials. Full account is taken of repeated roots, of arbitrary multiplicity, in the leading-order polynomial which determines a fractional-power expansion, namely the facet polynomial. For high multiplicity, the fractional powers are shown to have large denominators and contain irregularly spaced gaps. The orientation and methods of the paper are those of applications, but in a concluding section we draw attention to a more abstract approach, which is beyond the scope of the paper.

Item Type: Article
Official URL:
Additional Information: © 2021 The Authors
Divisions: Statistics
Subjects: H Social Sciences > HA Statistics
Date Deposited: 04 Feb 2022 18:30
Last Modified: 11 Feb 2022 08:54

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