The k-Beta Logarithmic Function: Theory, Fractional Derivative, and Spectral Numerical Method

A new generalization of the Logarithmic mean function and Euler’s Beta k-Logarithm function is proposed using the Mittag–Leffler k-function. We study their analytical properties, including functional relations, symmetry relation, inequalities, summation representations, and integral representations. Mellin transformations are established, and a generalized k-Beta Logarithmic distribution is presented along with its probabilistic applications. Furthermore, we introduce a novel k-Beta Logarithmic fractional derivative operator of Caputo type and develop a Legendre spectral collocation method with Chebyshev–Gauss–Lobatto nodes for the numerical solution of associated fractional differential equations. Rigorous error analysis in the weighted L2-norm is provided, establishing algebraic convergence for finite-regularity solutions and exponential convergence for analytic solutions. Numerical experiments confirm the theoretical convergence rates and demonstrate the efficiency and spectral accuracy of the proposed scheme.