Dual operators and Lagrange inversion in several variables

Abstract

The original Lagrange inversion formula, which gives explicitly the inverse under composition of a formal series, was obtained by Lagrange in 1770 through formal computations involving logarithms of inlmite products. See Lagrange [ 121. There exists an extensive literature on various versions of the inversion formula obtained by algebraic, analytic, and combinatorial methods. Comtet [3] gives a survey and an ample bibliography. During the last ten years there have appeared several papers on generalizations and new proofs of the inversion formula. In 1974 Abhyankar obtained an inversion formula using algebraic methods. See Bass, Connell, and Wright [ 11. Several authors have related Lagrange inversion with Rota’s Finite Operator Calculus [ 151, for example, Joni [8], Garsia and Joni [4], Hofhauer [7], and Roman and Rota [14]. Joyal and Labelle used a combinatorial theory of formal series to obtain inversion formulas (see [9-l I] ) and Viskov [ 161 used ideas from Lie algebra theory to prove formulas similar to Abhyankar’s. In the present paper we use operator methods to obtain inversion formulas in several variables that generalize the formulas of Abhyankar, Joni, and Viskov. We get our results studying an algebra with involution of linear operators on the algebra of formal Laurent series in several indeterminates. We also show that by means of an anti-isomorphism of operator algebras, which we call the operator Bore1 transform, the study of finite operator calculus can be reduced to the study of certain groups of operators on the algebra of formal Laurent series. Our approach may be considered as an algebraic analogue of the complex variable methods used by Good [5] to obtain the Lagrange inversion formula as a consequence of the change of variables theorem for the