arXiv subjects
Publications and source records attributed to Masao Ishikawa.
We present several generalizations of Cauchy's determinant and Schur's Pfaffian by considering matrices whose entries involve some generalized Vandermonde determinants. Special cases of our formulae include previuos formulae due to S.Okada and T. Sundquist. As an application, we give a relation for the Littlewood--Richardson coefficients involving a rectangular partition.
The initial purpose of this paper is to provide a combinatorial proof of the minor summation formula of Pfaffians based on the lattice path method. There we related Plücker relations with the minor summation formula of Pfaffians to simplify its proof by lattice paths. The second aim is to find a various applications of the minor summation formula. First we studied a variant of the Sundquist formula established in J. Alg. Combin. {\bf 5} (1996). Next we gave a simple proof of Kawanaka's formula concerning a $q$-series identity involving Schur functions in Osaka J. Math. {\bf 36} (1999). We also establish a certain identity similar to the Kawanaka formula, and give a combinatorial proof of the determinant version of his formula in Osaka J. Math. {\bf 38} (2001).
We present an identity which can be regarded as a Pfaffian-Hafnian analogue of Borchardt's identity and as a generalization of Schur's identity. We give a proof using the complex analysis.