Abstract: In this paper, we introduce and systematically study the Hyper G-Matrix Pair $$\left(A_n(s,t), B_n(s,t)\right)$$ where $$A_n(s,t)$$ is a Hankel matrix constructed from generalized harmonic numbers $$H_m(t,s)=\sum_{k=1}^{m}(k+s)^{-t}$$ and $$B_n(s,t)$$ is a binomial weight matrix generalizing the classical Hilbert matrix. We establish the fundamental duality relation $$A_n(s,t)^{-T}=D_1(s,t)B_n(s,t)D_2(s,t)$$ with explicit diagonal matrices $$D_1(s,t)$$ and $$D_2(s,t)$$. This relation extends the classical connection between the harmonic Hankel matrix and the Hilbert matrix for $$s=0,\ t=1$$. We provide explicit constructions, computational verifications, and structural properties of these matrix pairs. Connections to digamma functions, binomial coefficients, and Cauchy-type matrices are explored. The results unify and generalize several classical results in matrix analysis and special functions.

Keywords: Hyper G-Matrix, Hankel matrix, Hilbert matrix, generalized harmonic numbers, binomial coefficients, Cauchy matrix, digamma function, matrix duality, structured matrices, inverse formulas

Pages: 196-209