formal power series

• Polynomial extended to have an infinite number of terms
  • Corresponding to an infinite number sequence
    • $F = \sum_{i=0}^\infty f_i x^i$
    • At this point, the formal power series $F$ is the generating function of the sequence [$ {f_i}
  • The operation of taking the coefficients of the nth order terms from a formal power series F is written as $[x^n]F$.
• There is a convenient property inherited from the polynomial
  • Additions and subtractions
  • multiplication
    • $[x^n](F\times G) = \sum_{i+j=n} ([x^i]F \times [x^j] G)$

[Polynomial and Formal Power Series (2) Derivation of Solutions by Expression Transformation | maspy's HP https://maspypy.com/%e5%a4%9a%e9%a0%85%e5%bc%8f%e3%83%bb%e5%bd%a2%e5%bc%8f%e7%9a%84%e3%81%b9%e3 %81%8d%e7%b4%9a%e6%95%b0%ef%bc%88%ef%bc%92%ef%bc%89%e5%bc%8f%e5%a4%89%e5%bd%a2%e3%81%ab%e3%82%88%e3%82%8b%e8%a7%a3%e6%b3%95] - Infinite sum compression using the inverse of a formal power series - Use of Factorization - By allowing F to factorize, we can use binomial theorem for each of them. - $F^T = (A + B)^T (C + D)^T = (\sum_i \binom{T}{i}A^iB^{T-i} ) \times (\sum_j \binom{T}{j} C^jD^{T-j})$ - [exchange of product and sum - $F^T = (\sum_{i,j} \binom{T}{i}\binom{T}{j}A^iB^{T-i}C^jD^{T-j})$

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