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
SN - 0001-8708
VL - 44
SP - 180
EP - 205
JO - Advances in Mathematics
JF - Advances in Mathematics
IS - 2
ER -