Fractional power series and the method of dominant balances

C. J. Chapman and H. P. Wynn

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

Abstract: 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.

Keywords: Bell polynomials; Faà di Bruno's formula; Frobenius set; Newton polytope; Puiseux series; Sylvester set (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Pages: 19 pages
Date: 2021-02-17
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Published in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 17, February, 2021, 477(2246). ISSN: 1364-5021

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