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We show that the analytic bond-order potentials (BOPs) may be used to reproduce the density of states and energy of recursion-based methods for close-packed atomic configurations. In this way, we demonstrate that the analytic BOPs can efficiently recast the numerical bond-order potentials in a polynomial approximation. By introducing damping factors for the expansion coefficients in analogy to the kernel polynomial method, negative regions of the density of states are removed such that the analytic BOPs may be applied also to open systems with band gaps. By estimating higher moments from the termination of the Lanczos recursion chain, we, furthermore, achieve a faster convergence than the usual kernel polynomial method at a negligible additional computational cost.