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Through a brute-force approach to calculating the higher derivatives of thefalling factorial function, a number of interesting quantities were obtainedand analyzed. In particular, it was found that a quantity that can be describedas the "elementary symmetric harmonic sum" is a natural way to describe thesolutions to such higher derivatives. The relationship of this type ofgeneralized harmonic number to other quantities discussed in the literature,such as the r-Stirling numbers, was described in detail.

Higher Derivatives of the Falling Factorial and Related Generalizations of the Stirling and Harmonic Numbers