6 - Algebraic, D-Finite, and Noncommutative Generating Functions

Summary

Algebraic Generating Functions

In this chapter we will investigate two classes of generating functions called algebraic and D-finite generating functions. We will also briefly discuss the theory of noncommutative generating functions, especially their connection with rational and algebraic generating functions. The algebraic functions are a natural generalization of rational functions, while D-flnite functions are a natural generalization of algebraic functions. Thus we have the hierarchy

Various other classes could be added to the hierarchy, but the three classes of (6.1) seem the most useful for enumerative combinatorics.

Definition. Let K be a field. A formal power series η ∈ K [[x]] is said to be algebraic if there exist polynomials P₀(x),…, P_d(x) ∈ K[x], not all 0, such that

The smallest positive integer d for which (6.2) holds is called the degree of η.

Note that an algebraic series η has degree one if and only if η is rational. The set of all algebraic power series over K is denoted K_alg[[x]].

Example. Letη = ∑_n≥0x ⁿ. By Exercise 1.4(a) wehave(l-4x)η² - 1 = 0. Hence η is algebraic of degree one or two. If K has characteristic 2 then η = 1, which has degree one. Otherwise it is easy to see that η has degree two. Namely, if deg(η) = 1 then η = P(x)/Q(x) for some polynomials P(x), Q(x) ∈ K[x]. Thus

The degree (as a polynomial) of the left-hand side is odd while that of the right-hand side is even, a contradiction.