Solving the Matrix Equation $\sum _{\rho = 1}^r f_\rho (A)Xg_\rho (B) = C$

The equation of the title is attacked via Taylor's formula for matrix functions. This approach contrasts with that of Lancaster [1], who used methods of contour integration suggested by functional analysis. The paper is designed as a supplement to his, being organized around a number of generalizations of his theorems. Solutions of the target equation are expressed in terms of component matrices of A and B and in terms of contour integrals.