A Unified Proof of Three Combinatorial Identities Related to the Stirling Numbers of the Second Kind

Abstract: In the note, the authors give a unified proof of Identities~67, 84, and~85 in the monograph "M. Z. Spivey, The Art of Proving Binomial Identities, Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 2019; available online at https://doi.org/10.1201/9781351215824" and connect these three identities with a computing formula for the Stirling numbers of the second kind. Moreover, in terms of the notion of Qi's normalized remainders of the exponential and logarithmic functions, the authors reformulate the definitions of the Stirling numbers of the first and second kind and their generalizations by Howard in 1967 and 1980, Carlitz in 1980, and Broder in 1984.