The β-Extension of the Multivariable Lagrange Inversion Formula

Abstract

We show that Gessel's combinatorial proof of the multivariable Lagrange inversion formula can be given a ,β-extension, which generalizes Foata and Zeilberger's, β-extension of MacMahon's master theorem. Moreover, we show that there is no need to use Jacobi's identity in the derivation of the Lagrange formula. Finally, combining Gessel's method and ours, we obtain a new proof of Jacobi's identity.