Implicit Divided Differences, Little Schr\" oder Numbers, and Catalan Numbers
G Muntingh - arXiv preprint arXiv:1204.2709, 2012 - arxiv.org
arXiv preprint arXiv:1204.2709, 2012•arxiv.org
Under general conditions, the equation $ g (x, y)= 0$ implicitly defines $ y $ locally as a
function of $ x $. In this short note we study the combinatorial structure underlying a recently
discovered formula for the divided differences of $ y $ expressed in terms of bivariate
divided differences of $ g $, by analyzing the number of terms $ a_n $ in this formula. The
main result describes six equivalent characterizations of the sequence $\{a_n\} $.
function of $ x $. In this short note we study the combinatorial structure underlying a recently
discovered formula for the divided differences of $ y $ expressed in terms of bivariate
divided differences of $ g $, by analyzing the number of terms $ a_n $ in this formula. The
main result describes six equivalent characterizations of the sequence $\{a_n\} $.
Under general conditions, the equation implicitly defines locally as a function of . In this short note we study the combinatorial structure underlying a recently discovered formula for the divided differences of expressed in terms of bivariate divided differences of , by analyzing the number of terms in this formula. The main result describes six equivalent characterizations of the sequence .
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