Generalizations of Cauchy's determinant and Schur's Pfaffian

We present several identities of Cauchy-type determinants and Schur-type Pfaffians involving generalized Vandermonde determinants, which generalize Cauchy's determinant det(1/(xi+yj)) and Schur's Pfaffian Pf((xj−xi)/(xj+xi)). Some special cases of these identities are given by S. Okada and T. Sundquist. As an application, we give a relation for the Littlewood–Richardson coefficients involving a rectangular partition.