A Fractional Extension of the Hilbert Matrix and Its Spectral Properties

The spectral radius ρ(α, n) of a fractional extension of the n × n Hilbert matrix, defined through the Riemann–Liouville fractional integral, is investigated. The largest eigenvalue is computed numerically by the power iteration method for matrix dimensions 1 ≤ n ≤ 3.5 × 108, and its asymptotic behaviour is analysed using Pad´e approximants of increasing order.Closed-form expressions for the Frobenius norm are derived in terms of generalized Fox–Wright and Riemann zeta functions, providing analytical bounds for the spectral radius. The numerical and analytical results indicate that the spectral radius remains bounded as n → ∞ for α ≥ 1, whereas it diverges for α < 1.As a lowest-order approximation, the fractional Hilbert matrix is compared with a power-law Hilbert matrix, whose asymptotic spectral properties are investigated analytically. Finally, the construction is shown to extend naturally beyond the Riemann–Liouville setting, with tempered fractional integrals presented as an example of a broader class of fractional Hilbert matrices.