Hyperderivative power sums, Vandermonde matrices, and Carlitz multiplication coefficients

We investigate interconnected aspects of hyperderivatives polynomials over finite fields, q -th powers polynomials, and specializations Vandermonde matrices. construct formulas for Carlitz multiplication coefficients using symmetric we prove identities hyperderivative power sums in terms the inverse matrix. As an application these results give a new proof theorem Thakur on explicit Anderson's special log-algebraicity module. Furthermore, by combining Pellarin Perkins with techniques, obtain general case.