Periodic oscillations of coefficients of power series that satisfy functional equations

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Abstract

It is shown that the coefficients an of the power series f(z) = ∑ n=1 anzn which satisfies the functional equation f(z)=z+f(z2+z3). display periodic oscillations; an ∼ ( øn n) u(logn as n → ∞, where ø = (1 + 5 1 2) 2 and u(x) is a positive, nonconstant, continuous function which is periodic with period log(4 - ø). Similar results are obtained for a wide class of power series that satisfy similar functional equations. Power series of these types are of interest in combinatorics and computer science since they often represent generating functions. For example, the nth coefficient of the power series satisfying (*) enumerates 2, 3-trees with n leaves.

Original languageEnglish (US)
Pages (from-to)180-205
Number of pages26
JournalAdvances in Mathematics
Volume44
Issue number2
DOIs
StatePublished - May 1982

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