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Divided differences of inverse functions and partitions of a convex polygon


Authors: Michael S. Floater and Tom Lyche
Journal: Math. Comp. 77 (2008), 2295-2308
MSC (2000): Primary 05A17, 05A18, 26A06, 26A24, 41A05, 65D05
DOI: https://doi.org/10.1090/S0025-5718-08-02144-3
Published electronically: June 2, 2008
MathSciNet review: 2429886
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Abstract: We derive a formula for an $ n$-th order divided difference of the inverse of a function. The formula has a simple and surprising structure: it is a sum over partitions of a convex polygon with $ n+1$ vertices. The formula provides a numerically stable method of computing divided differences of $ k$-th roots. It also provides a new way of enumerating all partitions of a convex polygon of a certain type, i.e., with a specified number of triangles, quadrilaterals, and so on, which includes Catalan numbers as a special case.


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Additional Information

Michael S. Floater
Affiliation: Centre of Mathematics for Applications, Department of Informatics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway

Tom Lyche
Affiliation: Centre of Mathematics for Applications, Department of Informatics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway

DOI: https://doi.org/10.1090/S0025-5718-08-02144-3
Keywords: Divided differences, inverse functions, polygon partitions.
Received by editor(s): June 29, 2007
Published electronically: June 2, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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