Divided differences of inverse functions and partitions of a convex polygon

Floater, Michael; Lyche, Tom

doi:10.1090/S0025-5718-08-02144-3

Divided differences of inverse functions and partitions of a convex polygon
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by Michael S. Floater and Tom Lyche;

Math. Comp. 77 (2008), 2295-2308

DOI: https://doi.org/10.1090/S0025-5718-08-02144-3

Published electronically: June 2, 2008

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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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