Let $g(z) = a\prod_{i=1}^N (z-\lambda_i) \in \mathbb{Q}[z]$ be square-free. At each root $\lambda_i \in \mathbb{C}$, let $r_i$ denote the residue $\mathrm{Res}_{\lambda_i} 1/g(z)$. Let $I_g(z)$ denote the unique function of degree less than $N$ such that $I_g(\lambda_i) = r_i$ for all $i$. Note that $I_{ag}(z) = aI_{g}(z)$, so we may take $g(z) \in \mathbb{Z}[t]$ if so desired. For some basic examples, I've computed $I_g(z) = \frac{1}{5}(2z-1) \qquad \text{for} \qquad g(z) = z^2-z-1$ $I_g(z) = \frac{1}{22}(2z-1)(z-3) \qquad \text{for} \qquad g(z) = z^3 -z^2+z+1$
Has anyone seen reference to such a construction in the literature? My main questions are the following:
Q1: Do we have $I_g(z) \in \mathbb{Q}[z]$ for $g \in \mathbb{Q}[z]$?
Q2: Where $\Delta(g)$ denotes the discriminant of $g$, do we have $I_g(z) \in \frac{1}{\Delta(g)} \mathbb{Z}[z]$ for $g \in \mathbb{Z}[z]$? (A strengthening of Q1.)
Final note: the generalizations for $g$ not necessarily square-free are not difficult: take $I_g(z)$ be the unique function of degree less than $\deg g$ such that $I_g(\lambda_i)=r_i$, attained with multiplicity at least that of $\mathrm{ord}_{\lambda_i}g(z)$. We may also consider generalizations for polynomials with coefficients in a general number field.