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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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: http://dx.doi.org/10.1090/S0002-9947-97-01781-9
PII: S 0002-9947(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