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)
$ 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})
---
This page is auto-translated from /nishio/形式的べき級数. If you looks something interesting but the auto-translated English is not good enough to understand it, feel free to let me know at @nishio_en. I'm very happy to spread my thought to non-Japanese readers.