Truncated Plethystic Exponentials Preserve Power Sum Constraints

Abstract

Given an arbitrary sequence $(\hat{I}{\pm }_1,\dots ,\hat{I}{\pm }_n)\in {\mathbb{C}}^n$, we show that the degree-$n$ truncation of the formal exponential $\exp (-{\sum }_{k=1}^{\infty }\frac{\hat{I}{\pm }_k}{k}{x}^k)$ produces a polynomial whose roots $\ddot{I}{}_1,\dots ,\ddot{I}{}_n$ satisfy ${\sum }_{i=1}^n\ddot{I}{}_i^{-k}=\hat{I}{\pm }_k$ exactly for $k=1,\dots ,n$. This truncation-exactness property is an algebraic identity in the ring of formal power series, proved by coefficient matching. It defines a natural embedding of sequences into multisets of complex numbers and yields an $O(n^2)$ algorithm for computing the polynomial from the prescribed power sums. We apply the result to the polylogarithm family $\hat{I}{\pm }_k=k^{1-s}$, where the associated exponential $\exp (-{\mathrm{Li}}_s(x))$ produces factorial-integer coefficient sequences for $s\le 0$ and encodes values of the Riemann zeta function through ${\lim }_{n\to \infty }{P}_n^{(s)}(1)=\exp (-\hat{I}\P (s))$ for $\mathrm{Re}(s)>1$.

7 pages, 1 figure