A new extension of the Frank matrix and its properties

Motivated by the most recent literature, in this paper, we define a new generalization of the Frank matrix that we will call the geometric Frank matrix. In addition to considering some of its algebraic properties, we obtain its $LU$ factorization, its determinant, as well as a recurrence relation for its permanent. Upper bounds are set for the spectral norm. We also investigate similar properties for the Hadamard inverse of the geometric Frank matrix. Finally, we provide a MATLAB-R2023a code to facilitate the computation of the permanent of an arbitrary geometric Frank matrix.