Exploring Harmonic and Hyper-harmonic Narayana Numbers through Circulant Structures and Matrix Norms

This study investigates the structural and analytical properties of harmonic and hyper-harmonic Narayana numbers, extending classical results on harmonic and hyper-harmonic numbers. We establish novel combinatorial identities and derive closed-form expressions for and ​, which represent finite sums involving reciprocals of Narayana numbers. Furthermore, we examine the spectral and Euclidean norms of circulant and r-circulant matrices generated by these numbers, together with selected special matrix forms. Through this framework, several inequalities associated with matrix norms are obtained, providing deeper insight into the interplay between Narayana-type sequences and matrix analysis. The results contribute both to the combinatorial theory of special number sequences and to the normative analysis of structured matrices.