Reversion of power series and

the extended Raney coefficients

Authors:
Charles Ching-An Cheng, James H. McKay, Jacob Towber, Stuart Sui-Sheng Wang and David L. Wright

Journal:
Trans. Amer. Math. Soc. **349** (1997), 1769-1782

MSC (1991):
Primary 13F25, 05A15, 05C05, 13P99; Secondary 32A05

MathSciNet review:
1390972

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In direct as well as diagonal reversion of a system of power series, the reversion coefficients may be expressed as polynomials in the coefficients of the original power series. These polynomials have coefficients which are natural numbers (*Raney coefficients*). We provide a combinatorial interpretation for Raney coefficients. Specifically, each such coefficient counts a certain collection of ordered colored trees. We also provide a simple determinantal formula for Raney coefficients which involves multinomial coefficients.

**1.**Hyman Bass, Edwin H. Connell, and David Wright,*The Jacobian conjecture: reduction of degree and formal expansion of the inverse*, Bull. Amer. Math. Soc. (N.S.)**7**(1982), no. 2, 287–330. MR**663785**, 10.1090/S0273-0979-1982-15032-7**2.**Nicolas Bourbaki,*Elements of mathematics. Commutative algebra*, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR**0360549****3.**I. J. Good,*Generalizations to several variables of Lagrange’s expansion, with applications to stochastic processes*, Proc. Cambridge Philos. Soc.**56**(1960), 367–380. MR**0123021****4.**Mark Haiman and William Schmitt,*Incidence algebra antipodes and Lagrange inversion in one and several variables*, J. Combin. Theory Ser. A**50**(1989), no. 2, 172–185. MR**989192**, 10.1016/0097-3165(89)90013-7**5.**C. G. J. Jacobi,*De resolutione aequationum per series infinitas*, Journal für die reine und angewandte Mathematik**6**(1830), 257-286.**6.**Donald E. Knuth,*The art of computer programming. Vol. 1: Fundamental algorithms*, Second printing, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1969. MR**0286317**

Donald E. Knuth,*The art of computer programming*, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR**0378456**

D. E. Knuth,*The state of the art of computer programming*, Computer Science Department, School of Humanities and Sciences, Stanford University, Stanford, Calif.; distributed by National Technical Information Service, U. S. Department of Commerce, Springfield, Va., 1976. Errata to The art of computer programming, Vols. 1 and 2, Addison-Wesley, Reading, Mass., 1969; STAN-CS-76-551. MR**0478692****7.**George N. Raney,*Functional composition patterns and power series reversion*, Trans. Amer. Math. Soc.**94**(1960), 441–451. MR**0114765**, 10.1090/S0002-9947-1960-0114765-9**8.**J. Towber,*A combinatorial conjecture which implies the Jacobian conjecture*(to appear).**9.**Stuart Sui Sheng Wang,*A Jacobian criterion for separability*, J. Algebra**65**(1980), no. 2, 453–494. MR**585736**, 10.1016/0021-8693(80)90233-1**10.**David Wright,*The tree formulas for reversion of power series*, J. Pure Appl. Algebra**57**(1989), no. 2, 191–211. MR**985660**, 10.1016/0022-4049(89)90116-3

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
13F25,
05A15,
05C05,
13P99,
32A05

Retrieve articles in all journals with MSC (1991): 13F25, 05A15, 05C05, 13P99, 32A05

Additional Information

**Charles Ching-An Cheng**

Affiliation:
Department of Mathematical Sciences, Oakland University, Rochester, Michigan 48309-4401

Email:
cheng@vela.acs.oakland.edu

**James H. McKay**

Affiliation:
Department of Mathematical Sciences, Oakland University, Rochester, Michigan 48309-4401

Email:
mckay@vela.acs.oakland.edu

**Jacob Towber**

Affiliation:
Department of Mathematics, DePaul University, Chicago, Illinois 60614-3504

Email:
matjt@depaul.edu

**Stuart Sui-Sheng Wang**

Affiliation:
Department of Mathematical Sciences, Oakland University, Rochester, Michigan 48309-4401

Email:
swang@vela.acs.oakland.edu

**David L. Wright**

Affiliation:
Department of Mathematics, Washington University, St. Louis, Missouri 63130-4899

Email:
wright@einstein.wustl.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01781-9

Keywords:
Reversion of power series,
direct reversion,
diagonal reversion,
Jacobian conjecture,
colored trees,
colored forests,
inventory,
Raney coefficients,
Laurent series,
Jacobi's residue formula

Received by editor(s):
April 4, 1994

Additional Notes:
The third author was supported in part by the National Science Foundation under Grant DMS-9012210.\endgraf The fifth author was supported in part by the National Security Agency under Grant MDA-904-89-H-2049

Article copyright:
© Copyright 1997
American Mathematical Society