Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: A Multivariate Lagrange Inversion Formula for Asymptotic. | A Multivariate Lagrange Inversion Formula for Asymptotic Calculations Edward A. Bender Department of Mathematics University of California San Diego La Jolla CA 92093-0112 USA ebender@ L. Bruce Richmond Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario N2L 3G1 Canada lbrichmond@ Submitted March 3 1998 Accepted June 30 1998 Abstract The determinant that is present in traditional formulations of multivariate Lagrange inversion causes difficulties when one attempts to obtain asymptotic information. We obtain an alternate formulation as a sum of terms thereby avoiding this difficulty. 1991 AMS Classification Number. Primary 05A15 Secondary 05C05 40E99 THE ELECTRONIC JOURNAL OF COMBINATORICS 5 1998 R33 2 1. Introduction Many researchers have studied the Lagrange inversion formula obtaining a variety of proofs and extensions. Gessel 4 has collected an extensive set of references. For more recent results see Haiman and Schmitt 6 Goulden and Kulkarni 5 and Section of Bergeron Labelle and Leroux 3 . Let boldface letters denote vectors and let a vector to a vector power be the product of componentwise exponentiation as in xn x 1 xn. Let xn h x denote the coefficient of xn in h x . Let ai jII denote the determinant of the d X d matrix with entries ai j. A traditional formulation of multivariate Lagrange inversion is Theorem 1. Suppose that g x f1 x fd x are formal power series in x such that fi 0 0 for 1 i d. Then the set of equations wi tifi w for 1 i d uniquely determine the wi as formal power series in t and tn g w t xn g x f x n . _ xi @fj x ij fj x @xi 1 where si j is the Kronecker delta. If one attempts to use this formula to estimate tn g w t by steepest descent or stationary phase one hnds that the determinant vanishes near the point where the integrand is maximized and this can lead to difficulties as min ni 1. We derive an alternate formulation of 1 which avoids this problem. In 2 we apply the .
