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. - $F^T=(A+B{)}^T(C+D{)}^T=({\sum }_i(\genfrac{}{}{0ex}{}{T}{i})A^i{B}^{T-i})\times ({\sum }_j(\genfrac{}{}{0ex}{}{T}{j})C^j{D}^{T-j})$ - [exchange of product and sum - $F^T=({\sum }_{i,j}(\genfrac{}{}{0ex}{}{T}{i}\left)\right(\genfrac{}{}{0ex}{}{T}{j})A^i{B}^{T-i}{C}^j{D}^{T-j})$

- [[Derivation of dp transition by cumulative sum]]
- [[Derivation of DP to be returned]]
- [[Application of the Law of Exchange and Repeated Squares]]

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