Analytic Combinatorics by Mishna Marni;

Author:Mishna, Marni;
Language: eng
Format: epub
Publisher: CRC Press LLC
Published: 2019-11-04T00:00:00+00:00

FIGURE 5.1

Venn diagram of function classes.

5.1 Rational Functions

A series is rational over K if it satisfies an equation of the form for polynomials . We write . It is furthermore qualified to be -rational if it is contained in the smallest set of rational functions containing which is closed under addition, multiplication and quasi-inverse.

Theorem 4.10 contained very precise information about the location of the singularities, and the form of the asymptotics of rational function coefficients. It turns out that -rational functions have additional analytic structure that can be exploited.

Theorem 5.1 (Berstel, 1971) If F(z) is -rational with radius of convergence ρ then ρ is a pole of F(z). Furthermore if ρ′ is a pole of F(z) with , then ρ′/ρ is a (integral) root of unity.

If F(z) is -rational then it can be expressed as a sum of rational functions with a single (positive) pole on the circle of convergence.

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