formal power series

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formal power series

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Noun

formal power series (plural formal power series)

1. (mathematics, algebra) Any finite or infinite series of the form ${\displaystyle \textstyle a_{0}+a_{1}x+a_{2}x^{2}\dots =\sum _{i}{a_{i}x^{i}}}$, where the a_i are numbers, but it is understood that no value is assigned to x.
   • 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. ${\displaystyle r\in \mathbb {Q} \left\langle \!\langle z\right\rangle \!\rangle }$ 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.

   • 2011, Nicholas Loehr, Bijective Combinatorics‎, Taylor & Francis (CRC Press), page 243:

     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.

   Synonym: formal series

Usage notes

The formal power series concept can be considered either as an extension of the polynomial to a possibly infinite number of terms or as a power series in which the variable is not assigned any value.

Addition and multiplication operations are defined, allowing formal power series to be studied in the context of group theory. The set of all formal power series in X with coefficients in a commutative ring R forms another ring called the ring of formal power series in the variable X over R, commonly written R[[X]].

See also

• formal Laurent series
• Lagrange inversion theorem
• Lagrange-Bürmann formula

Further reading

• Formal Power Series, Eric W. Weissman, MathWorld