• i. Fast Fourier Transform: Complex FFT with packed real convolution, exact rounding and arbitrary-modulus splitting, the convolution toolkit, Karatsuba and big-number multiplication.
• ii. Number-Theoretic Transform: NTT over friendly primes, the shared Poly core (convolution, polyMul) and three-prime CRT convolution for any modulus.
• iii. Convolution Variants: 2D, digit-wise (base-$k$ XOR/OR/AND), chirp-z/Bluestein, $(\min ,+)$ with convexity, and ${\mathbb{F}}_{2^{64}}$ arithmetic.
• iv. Online Convolution: CDQ and fully relaxed online convolution for self-referential series, with rooted-tree counting.
• v. Power Series Operations: Newton iteration: inverse, log, exp, sqrt, power and division of power series, synthetic division, product trees, EGF dictionary.
• vi. Multipoint Evaluation and Interpolation: Subproduct tree evaluation and interpolation, $O(n)$ Lagrange, power sums, sample shifting, $n!\mathrm{mod}p$ in $\tilde{O}(\sqrt{n})$, Newton basis.
• vii. Composition and Taylor Shift: Composition (Brent–Kung, Kinoshita–Li), power projection, reversion via Lagrange inversion, Taylor shift and binomial sums.
• viii. Polynomial GCD and Resultant: Half-gcd, inverse modulo a polynomial, resultant and discriminant.
• ix. Counting with Power Series: Stirling/Bell/Bernoulli/partition series, Euler transform and subset-sum counts, sparse series in $O(nk)$, sums of exponential times polynomial.
• x. Linear Recurrences: Kitamasa, Bostan–Mori, Berlekamp–Massey (with Massey’s lemma), Reeds–Sloane for composite moduli, P-recursive sequences.
• xi. Polynomial Factorization: Square-free, distinct- and equal-degree factorisation (Cantor–Zassenhaus), roots mod $p$, Rabin test, Berlekamp, cyclotomic polynomials.