Formal Power Series

Abstract

We begin here the subject of formal power series, objects of the form $\displaystyle \sum _{n=0}^{\infty }a_nX^n$ ($a_n\in \mathbb R$ or $\mathbb C)$ which can be thought as a generalization of polynomials. We focus here on their algebraic properties and basic applications to combinatorics. The reader must not be confused by the many technical, though simple, details that are needed in a book to justify rigorously every step. In Chap. 3, we have learned how to count sequences and collections with occupancy constraints: the number of possible codes of 10 digits that use only four 1’s, five 2’s and one 3 is easily obtained: $\dfrac{10!}{4!5!}$. What about counting the possible codes of 10 digits that use an even number of 1’s, an odd number of 2’s and any number of 3’s? What about the validity of the “Latin teacher’s random choice” which selects the students to test by opening randomly a book and summing up the digits of the page? Many counting problems can be solved using the formal power series! These are extremely useful in studying recurrences, notable sequences, probability and many other arguments. Moreover, they constitute a rich and interesting environment in their own right. The chapter ends with a combinatorial proof of the celebrated Euler pentagonal theorem.