Using power series may enhance the readability and interpretability of expressions

[log2cn]

In Section 2, we have $T_{\omega^k}(s) = \frac{f(s)}{s-\omega^k} = s^{-1} f(s) + s^{-2} f(s) \omega^k + ... + s^{-m} f(s) (\omega^k)^{m-1}$, where we ignore the negative exponents of $s$. We denote $H(x) = s^{-1} f(s) + s^{-2} f(s) x + ... + s^{-m} f(s) x^{m-1}$. Thus, we can express $T_{\omega^k}(s)$ as $H(\omega^k)$. This makes it clear that we can compute $T_{\omega^k}(s)$ by applying FFT on the coefficients of $H$.

In Section 3, this writing style also applicable. We denote $\varphi = \omega^l$. Then, we have $T_{\omega^k}(s) = \frac{f(s)}{s^l-(\omega^l)^k} = \frac{f(s)}{s^l-\varphi^k} = s^{-l} f(s) + s^{-2l} f(s) \varphi^k + ... + s^{-ml} f(s) (\varphi^k)^{m-1}$. We again define $H(x) = s^{-l} f(s) + s^{-2l} f(s) x + ... + s^{-ml} f(s) x^{m-1}$, allowing us to express $T_{\omega^k}(s) = H(\varphi^k)$. This also makes it clear that we can compute $T_{\omega^k}(s)$ by applying FFT on the coefficients of $H$. We can compute the coefficients of $H$ by grouping them according to the residue modulo $l$, as illustrated in the following diagram (where $l=2$).