A note on an identity of Jacobi's

Abstract

where Ar is the r' A complete homogeneous symmetric function in a set of n arguments, is equal to the quotient of a particular pair of alternants was shown essentially by Jacobi in 1841 and by Trudi in 1864. The present note exhibits this well-known relation, (3), as the immediate consequence of a simple matrix equality. The symmetric functions hr are connected with the elementary symmetric functions ar in the same n arguments a, /?, . . . , K by the Wronski relations Og/tj — axh0 = 0, a0h2 — a1h1 + ath0 = 0, aohs — ajiz + «2^i — 3^o = 0>