Abstract
We present an algebraic theory of divided differences which includes confluent differences, interpolation formulas, Liebniz's rule, the chain rule, and Lagrange inversion. Our approach uses only basic linear algebra. We also show that the general results about divided differences yield interesting combinatorial identities when we consider some suitable particular cases. For example, the chain rule gives us generalizations of the identity used by Good in his famous proof of Dyson's conjecture. We also obtain identities involving binomial coefficients, Stirling numbers, Gaussian coefficients, and harmonic numbers.
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