Generating Functions for Divisor Sums and Totative Sums Arising from Combinatorial Simsek Numbers

Abstract

The main objective of this paper is to introduce and investigate new number families derived from finite sums running over divisors and totatives and containing higher powers of binomial coefficients. Especially, by making decomposition on the generating functions for a kind of combinatorial number families recently introduced by Simsek [29], we also construct generating functions for the newly introduced number families. For symbolic computation of the newly introduced number families and their generating functions, we also give computational implementations in Wolfram language. By these implementations, some tables of both these number families and their generating functions have been presented for some arbitrarily chosen special cases. Additionally, we provide some applications regarding the Thacker's (totient) function. Especially, by making summation on all totatives, we investigate some special finite sums containing both the Thacker's (totient) function and higher powers of binomial coefficients. By this investigation, some of the questions regarding these finite sums partially answered accompanied by some remarks. Furthermore, we propose an open question regarding a potential relation between one these number families and a formula involving the Möbius function. Finally, the paper have been concluded by providing an overview on the results of this paper and their potential usage areas, and by making suggestions regarding future studies able to be made.