Continued fraction representations of the generalized operator entropy

The direct calculation of the Generalized operator entropy proves difficult by the appearance of rational exponents of matrices. The main motivation of this work is to overcome these difficulties and to present a practical and efficient method for this calculation using its representation by the matrix continued fraction. At the end of our paper, we deduce a continued fraction expansion of the Bregman operator divergence. Some numerical examples illutrating the theoretical result are discussed.