J Austral Math Soc Ser A 56 pp131--143, 1994.

How to Obtain an Asymptotic Expansion of a Aequence from an Analytic Identity Satisfied by its Generating Function

J. M. Plotkin and John W. Rosenthal

Abstract

Let f_n be a sequence of nonnegative integers and let f(x) : = å_{n ³ 0} f_nxⁿ be its generating function. Assume f(x) has the following properties: it has radius of convergence r, 0 < r < 1, with its only singularity on the circle of convergence at x = r and f(r) = s; y = f(x) satisfies an analytic identity F(x, y) = 0 near (r, s); and for some k ³ 2 F_0,j = 0, 0 £ j < k, F_0,k ¹ 0 where F_i,j is the value at (r, s) of the i^th partial derivative with respect to x and the j^th partial derivative with respect to y of F. These assumptions form the basis of what we call the typical and general cases. In both cases we show how to obtain an asymptotic expansion of f_n. We apply our technique to produce several terms in the asymptotic expansion of combinatorial sequences for which previously only the first term was known.

1991 AMS Subject Classification: primary 05A15; secondary 05C30

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Authors

J. M. Plotkin

Department of Mathematics, Michigan Stae University, East Lansing, Michigan 48824.

John W. Rosenthal

Department of Mathematics and Computer Science, Ithaca College, Ithaca, New York 14850.