An algebraic approach to the generalized Stirling numbers is presented, leading to a unified interpretation for important combinatorial functions such as the binomials, Stirling numbers, and Gaussian polynomials. Let \(G\) be the group of all infinite lower triangular matrices \(A\) over a field \(K\) of characteristic \(0\) for which \(A(n,m)= 1\), for all \(n\in N_0\). To define a mapping \(\varphi\) from the set \(M\) of all coefficient functions \(a: N_0\times N_0\to K\) to \(G\), the following recurrence relation is used: \(A(0,l)= \delta(0,l)\) and \(A(n,l)= a_{n-1,l} A(n-1,l)+ A(n- 1,l-1)\). By restricting \(\varphi\) to appropriate subsets \(N\) and \(G\) of \(M\), the main result of this paper is: \(\varphi(a+ b)= \varphi(a)\cdot\varphi(b)\) for all \(a\in N\) and \(b\in G\). A Chu-Vandermonde type convolution formula for the generalized Stirling numbers is obtained by introducing arbitrary boundary conditions. Finally the problem of interpolating between sequences of polynomials with persistent roots is investigated, and a refinement of the earlier analysis of the structure of the connection constants is presented by using the main result of this paper.