formal power series

Definitions

Any finite or infinite series of the form a_0+a_1x+a_2x²…=∑ᵢa_ixⁱ, where the aᵢ are numbers, but it is understood that no value is assigned to x.

Examples

“1995 Alberto Bertoni, Massimiliano Goldwurm, Giancarlo Mauri, Nicoletta Sabatini, Chapter 5: Counting Techniques for Inclusion, Equivalence, and Membership Problems, Volker Diekert, Grzegorz Rozenberg, The Book of Traces, World Scientific, page 138, Moreover, it is useful to observe⟳ that in this case the definition of rational formal power series can be simplified: a f.p.s. r∈ℚ⟨!⟨z⟩!⟩ is rational if and only if r is the quotient of two polynomials.”

“1997, Greg Marks, Direct⟳ Product and Power Series Formations over 2-Primal Rings, Surender Kumar Jain, S. Tariq Rizvi, Advances in Ring⟳ Theory, Springer, page 239, We also show⟳ that the ring⟳ of formal power series over a 2-primal ring⟳ (or even a ring⟳ satisfying (PS I)) need⟳ not be 2-primal.”

“This chapter gives a rigorous development of the algebraic properties of formal power series. Our goal is to extend⟳ the familiar operations on polynomial functions (like⟳ addition, multiplication, composition, and differentiation) to the setting of formal power series. In certain situations we will even be able to define⟳ infinite sums and products of formal power series.[…]In combinatorics, it usually suffices to consider⟳ formal power series whose coefficients are integers, rational numbers, or complex numbers.”

CEFR level