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Some asymptotic methods in combinatorics

Part of: Combinatorics

Published online by Cambridge University Press:  09 April 2009

J. M. Plotkin  and
John Rosenthal
J. M. Plotkin
Affiliation:
Michigan State UniversityEast Lansing 48824, USA
John Rosenthal
Affiliation:
Ithaca CollegeIthaca 14850, USA
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Abstract

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Let 〈fn≧0 be nonnegative real numbers with generating function f(x) = Σfnxn. Assume f(x) has the following properties: it has a finite nonzero radius of convergence x0 with its only singularity on the circle of convergence at x = x0 and f(x0) converges to y0; y = f(x) satisfies an analytic identity F(x, y) = 0 near (x0, y0); Fy(l) (x0, y0)= 0, 0 ≦ i < k and Fy(k) (x0, y0) ≠ 0. There are constants γ, a positive rational, and c such that fn~cx0−n n−(1 +ggr;). Furthermore, we show (i) in all cases how to determine γ and c from f(x) and (ii) in certain cases how to determine them from F(x, y).

MSC classification

Secondary: 05A15: Exact enumeration problems, generating functions

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 4 , December 1979 , pp. 452 - 460
Copyright
Copyright © Australian Mathematical Society 1979

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