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

2014/01/13 by Steven S.S. Poon, Steven S. Poon, Poon, Steven S.
Decision Sciences · Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Orbital Angular Momentum in Optics #Scientific Measurement and Uncertainty Evaluation #math.CO

paper · pdf · doi:10.48550/arxiv.1401.2737

34 pages

arxiv created 2014/01/13 · openalex publication_date 2014/01/13 · arxiv updated 2014/01/14 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28

Abstract

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

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