Generating functions for divisor sums and totative sums arising from combinatorial Simsek numbers
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Tarih
2024
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Univ Nis, Fac Sci Math
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
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 the 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. In particular, by making summation on all totatives of a positive integer, 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 problems regarding these finite sums have been partially answered accompanied by some remarks. Furthermore, we propose an open problem regarding a potential relation between one of these number families and a formula involving the Mo center dot bius 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.
Açıklama
Anahtar Kelimeler
. Binomial coefficients, Computational implementations, Divisor sums, Generalized Franel numbers, Generating func-tion, Special numbers, Combinatorial Simsek numbers, Number-theoretic (arithmetical) functions, Mo center dot bius function, Euler's totient function, Thacker's (totient) function, Totatives
Kaynak
Filomat
WoS Q Değeri
Q2
Scopus Q Değeri
Q2
Cilt
38
Sayı
5
