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

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

MathSciNet review:
1390972

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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.

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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