Neutron Fluctuations

APPENDIX A

ELEMENTS OF THE THEORY OF GENERATING FUNCTIONS

Contents

A.1. Basic Properties

A.2. On the Roots of Equation g(x) = x, 0 ≤ x ≤ 1

A.3. A Useful Inequality

A.4. Abel Theorem for Moments

A.5. Series Expansion Theorem

A.6. An Important Theorem

In this chapter, we summarise some of the elementary properties of the generating functions that play an exclusively important role in the theory and applications of branching processes. Roughly speaking, a generating function is a polynomial, a power series, whose exponents are non-negative integers and whose coefficients are probabilities of discrete random variables which assume non-negative integer values.

A.1 BASIC PROPERTIES

Define the discrete random variables x_n, n = 1, 2, …, taking ...