On dualistic structure involving Shannon transform and integrated R-transform

We consider a problem of evaluating average of a certain scalar function involving a random matrix in the large- dimension limit, the solution of which is given in terms of the integrated R-transform of the limiting eigenvalue distribution of the random matrix. This problem therefore serves as an example in which not the functional form but the values of the R-transform plays a significant role, which can be regarded as making this problem unique in the context of application of random matrix and free probability theories. We furthermore discuss a dualistic structure with Legendre transformation involving the Shannon transform and the integrated R-transform, which underlies the problem.