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Divided Differences and Combinatorial Identities
Author(s) -
VerdeStar Luis
Publication year - 1991
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1991853215
Subject(s) - binomial coefficient , mathematics , stirling numbers of the second kind , stirling number , conjecture , gaussian binomial coefficient , algebraic number , combinatorial proof , identity (music) , binomial (polynomial) , stirling numbers of the first kind , harmonic number , inversion (geology) , gaussian , lattice (music) , algebra over a field , interpolation (computer graphics) , pure mathematics , combinatorics , mathematical analysis , negative binomial distribution , frame (networking) , statistics , computer science , riemann zeta function , paleontology , telecommunications , physics , structural basin , quantum mechanics , acoustics , poisson distribution , biology
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.