Akademik Veri Yönetim Sistemi
PHYSICA SCRIPTA, cilt.100, sa.10, 2025 (SCI-Expanded, Scopus)
In this paper, we introduce three new generalizations of harmonic numbers. Connections are established by considering the rho-th derivative of the rising factorial for rho is an element of Z+ , leading to identities involving these generalized harmonic numbers. Using these connections together with the first kind Stirling numbers, properties of the derivative operator, and the Sigma.m Mathematica package, several identities and summation formulas are derived. Furthermore, we identify a relation between the rho-th derivative of the Gamma function and complete Bell polynomials. By combining this relation with Gauss' hypergeometric theorem, we obtain important series involving some generalized harmonic numbers, the Gamma function, and complete Bell polynomials. Then, these series are specialized for a parameter & varsigma;is an element of R+ using the digamma function psi and fundamental mathematical operations, yielding explicit results. Finally, as an application of these results, further series expansions are presented.