Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems

Clarke, Francis; Hunton, John; Ray, Nigel

doi:10.5802/aif.1824

Cited by 1 publication

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“…In combinatorics it is worth noticing the interrelation of formal group theory with umbral calculus [6,33]. As for applications in other areas of mathematics, the theory of formal groups has played, in fact, a sort of unifying role.…”

Section: Appendixmentioning

confidence: 99%

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González

2003

Journal of Algebraic Combinatorics

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The 1-dimensional universal formal group law is a power series (in two variables and with coefficients in Lazard's ring) carrying a lot of geometrical and algebraic properties. For a prime p, we study the corresponding " p-localized" formal group law through its associated p k-series, [ p k ](x) = s≥0 a k,s x s(p−1)+1-the p k-fold iterated formal sum of a variable x. The coefficients a k,s lie in the Brown-Peterson ring BP * = Z (p) [v 1 , v 2 ,. . .] and we describe part of their structure as polynomials in the variables v i with p-local coefficients. This is achieved by introducing a family of filtrations {W ϕ } ϕ≥1 in BP * and studying the value of a k,s in each of the associated (bi)graded rings BP * /W ϕ. This allows us to identify, among monomials in a k,s of minimal W ϕ-filtration (1 ≤ ϕ ≤ k), an explicit monomial m ϕ,k,s carrying the lowest possible p-divisibility. The p-local coefficient of m ϕ,k,s is described as a Stirling-type number of the second kind and its actual value is computed up to p-local units. It turns out that m k,k,s not only carries the lowest W k-filtration but, more importantly, the lowest p-divisibility among all other monomials in a k,s. In particular, we obtain a complete description of the p-divisibility properties of each a k,s .

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Section: Appendixmentioning

confidence: 99%

Untitled

González

2003

Journal of Algebraic Combinatorics

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show abstract

Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems

Cited by 1 publication

References 24 publications

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