TY - JOUR

T1 - Periodic oscillations of coefficients of power series that satisfy functional equations

AU - Odlyzko, A. M.

PY - 1982/5

Y1 - 1982/5

N2 - 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.

AB - 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.

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U2 - 10.1016/0001-8708(82)90005-6

DO - 10.1016/0001-8708(82)90005-6

M3 - Article

AN - SCOPUS:0001725943

VL - 44

SP - 180

EP - 205

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 2

ER -