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We consider the lower-triangular matrix of generating polynomials thatenumerate $k$-component forests of rooted trees on the vertex set $[n]$according to the number of improper edges (generalizations of the Ramanujanpolynomials). We show that this matrix is coefficientwise totally positive andthat the sequence of its row-generating polynomials is coefficientwiseHankel-totally positive. More generally, we define the generic rooted-forestpolynomials by introducing also a weight $m! \, \phi_m$ for each vertex with$m$ proper children. We show that if the weight sequence $\phi$ isToeplitz-totally positive, then the two foregoing total-positivity resultscontinue to hold. Our proofs use production matrices and exponential Riordanarrays.

Total positivity of some polynomial matrices that enumerate labeled trees and forests, I. Forests of rooted labeled trees