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In the paper, the authors study properties of three functions relating to the exponential function and the existence of partitions of unity, including accurate and explicit computation of their derivatives, analyticity, complete monotonicity, logarithmically complete monotonicity, absolute monotonicity, and the like. Finally, the authors pose an open problem.doi:10.12816/0006128 fatcat:mr5z65so5bds7jsabwefg3vuau