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formal power series

出典:『Wiktionary』 (2026/05/04 19:35 UTC 版)

名詞

formal power series (plural **formal power series)

  1. (mathematics, algebra) Any finite or infinite series of the form a 0 + a 1 x + a 2 x 2 ⋯ = ∑ i a i x i {\displaystyle \textstyle a_{0}+a_{1}x+a_{2}x^{2}\dots =\sum _{i}{a_{i}x^{i}}} {\displaystyle \textstyle a_{0}+a_{1}x+a_{2}x^{2}\dots =\sum _{i}{a_{i}x^{i}}}, where the ai 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. r ∈ Q ⟨ ⟨ z ⟩ ⟩ {\displaystyle r\in \mathbb {Q} \left\langle \!\langle z\right\rangle \!\rangle } {\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.
      Synonym: formal series

参考

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