ASYMPTOTIC ENUMERATION METHODS

By X. LUURO

CONTENTS INTRODUCTION NOTATION IDENTITIES, INDE?NITE SUMMATIONS, AND RELATED APPROACHES BASIC ESTIMATES: FACTORIALS AND BINOMIAL COE?CIENTS ESTIMATES OF SUMS AND OTHER BASIC TECHNIQUES SUMS OF POSITIVE TERMS ALTERNATING SUMS AND THE PRINCIPLE OF INCLUSION-EXCLUSION EULER-MACLAURIN AND POISSON SUMMATION FORMULAS BOOTSTRAPPING AND OTHER BASIC METHODS ESTIMATION OF INTEGRALS GENERATING FUNCTIONS A BRIEF OVERVIEW COMPOSITION AND INVERSION OF POWER SERIES DI?ERENTIABLY ?NITE POWER SERIES UNIMODALITY AND LOG-CONCAVITY MOMENTS AND DISTRIBUTIONS FORMAL POWER SERIES ELEMENTARY ESTIMATES FOR CONVERGENT GENERATING FUNCTIONS SIMPLE UPPER AND LOWER BOUNDS TAUBERIAN THEOREMS RECURRENCES LINEAR RECURRENCES WITH CONSTANT COE?CIENTS LINEAR RECURRENCES WITH VARYING COE?CIENTS LINEAR RECURRENCES IN SEVERAL VARIABLES NONLINEAR RECURRENCES QUASI-LINEAR RECURRENCES 10 ANALYTIC GENERATING FUNCTIONS 10 1 INTRODUCTION AND GENERAL ESTIMATES 10 2 SUBTRACTION OF SINGULARITIES 10 3 THE RESIDUE THEOREM AND SUMS AS INTEGRALS 10 4 LOCATION OF SINGULARITIES, ROUCH´E’S THEOREM, AND UNIMODALITY 10 5 IMPLICIT FUNCTIONS 11 SMALL SINGULARITIES OF ANALYTIC FUNCTIONS 11 1 TRANSFER THEOREMS 11 2 DARBOUX’S THEOREM AND OTHER METHODS 12 LARGE SINGULARITIES OF ANALYTIC FUNCTIONS 12 1 THE SADDLE POINT METHOD 12 2 ADMISSIBLE FUNCTIONS 12 3 OTHER SADDLE POINT APPLICATIONS 12 4 THE CIRCLE METHOD AND OTHER TECHNIQUES 13 MULTIVARIATE GENERATING FUNCTIONS 14 MELLIN AND OTHER INTEGRAL TRANSFORMS 15 FUNCTIONAL EQUATIONS, RECURRENCES, AND COMBINATIONS OF METHODS 15 1 IMPLICIT FUNCTIONS, GRAPHICAL ENUMERATION, AND RELATED TOPICS 15 2 NONLINEAR ITERATION AND TREE PARAMETERS 15 3 DI?ERENTIAL AND INTEGRAL EQUATIONS 15 4 FUNCTIONAL EQUATIONS 16 OTHER METHODS 16 1 PERMANENTS 16 2 PROBABILITY THEORY AND BRANCHING PROCESS METHODS 16 3 STATISTICAL PHYSICS 16 4 CLASSICAL APPLIED MATHEMATICS 17 ALGORITHMIC AND AUTOMATED ASYMPTOTICS 18 GUIDE TO THE LITERATURE