×

Extensions of umbral calculus. II: Double delta operators, Leibniz extensions and Hattori-Stong theorems. (English) Zbl 0962.05012

Summary: “We continue our programme of extending the Roman-Rota umbral calculus to the setting of delta operators over a graded ring \(E_{*}\) with a view to applications in algebraic topology and the theory of formal group laws. We concentrate on the situation where \(E_{*}\) is free of additive torsion, in which context the central issues are number-theoretic questions of divisibility. We study polynomial algebras which admit the action of two delta operators linked by an invertible power series, and make related constructions motivated by the Hattori-Stong theorem of algebraic topology. Our treatment is couched purely in terms of the umbral calculus, but inspires novel topological applications. In particular we obtain a generalised form of the Hattori-Stong theorem.”
For Part I see N. Ray [Adv. Math. 61, 49-100 (1986; Zbl 0631.05002)].

MSC:

05A40 Umbral calculus
55N22 Bordism and cobordism theories and formal group laws in algebraic topology

Citations:

Zbl 0631.05002
PDFBibTeX XMLCite
Full Text: DOI Numdam Numdam EuDML

References:

[1] On Chern characters and the structure of the unitary group, Proc. Cambridge Philos. Soc., 57, 189-199 (1961) · Zbl 0103.16001 · doi:10.1017/S0305004100035052
[2] Stable homotopy and generalised homology (1974) · Zbl 0309.55016
[3] Combinatorial and arithmetic identities based on formal group laws, Algebraic topology, Barcelona 1986, 1298, 17-34 (1987) · Zbl 0666.14019
[4] Some properties of Hurwitz series, Duke Math. J, 16, 285-295 (1949) · Zbl 0041.17401 · doi:10.1215/S0012-7094-49-01627-0
[5] The universal von Staudt theorems, Trans. Amer. Math. Soc., 315, 591-603 (1989) · Zbl 0683.10013 · doi:10.1090/S0002-9947-1989-0986687-3
[6] Analyse Combinatoire (1970) · Zbl 0221.05001
[7] Advanced Combinatorics (1974) · Zbl 0283.05001
[8] Sur les produits tensoriels, Ann. Sci. École Norm. Sup, Série 3, 64, 101-117 (1947) · Zbl 0033.24801
[9] Formal groups, 74 (1968) · Zbl 0177.04801
[10] Abelian groups, Hungarian Acad. Sci., Budapest (1958) · Zbl 0091.02704
[11] Die Gruppe der \(p^n\)-primären Zahlen für einen Primteiler \(\mathfrak{p}\) von \(p\), J. Reine Angew. Math., 176, 174-183 (1936) · JFM 62.1115.01
[12] Integral characteristic numbers for weakly almost complex manifolds, Topology, 5, 259-280 (1966) · Zbl 0146.19401 · doi:10.1016/0040-9383(66)90010-3
[13] Formal groups and applications (1978) · Zbl 0454.14020
[14] \(BP_*(BP)\) and typical formal groups, Osaka J. Math., 12, 357-363 (1975) · Zbl 0311.55003
[15] Homological properties of comodules over \(MU_*(MU)\) and \(BP_*(BP)\), Amer. J. Math., 98, 591-610 (1976) · Zbl 0355.55007 · doi:10.2307/2373808
[16] Supersingular elliptic curves and congruences for Legendre polynomials, Elliptic curves and modular forms in algebraic topology, Princeton (1986), 1326, 69-83 (1988) · Zbl 0655.14018
[17] The topological \(q\)-expansion principle, Topology, 38, 387-425 (1999) · Zbl 0924.55004 · doi:10.1016/S0040-9383(98)00019-6
[18] Morava stabilizer algebras and the localisation of Novikov’s \(E_2\)-term, Duke Math. J., 44, 433-447 (1977) · Zbl 0358.55019 · doi:10.1215/S0012-7094-77-04420-9
[19] On the cobordism ring \(\Omega^*\) and a complex analogue (part I), Amer. J. Math., 82, 505-521 (1960) · Zbl 0095.16702
[20] On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc., 75, 1293-1298 (1969) · Zbl 0199.26705 · doi:10.1090/S0002-9904-1969-12401-8
[21] Extensions of umbral calculus: penumbral coalgebras and generalised Bernoulli numbers, Adv. Math., 61, 49-100 (1986) · Zbl 0631.05002 · doi:10.1016/0001-8708(86)90065-4
[22] Symbolic calculus: a 19th century approach to \(MU\) and \(BP\), Homotopy theory, Durham (1985), 195-238 (1987) · Zbl 0651.55004
[23] Stirling and Bernoulli numbers for complex oriented homology theories, Algebraic topology, Arcata, CA, (1986), 1370, 362-373 (1989) · Zbl 0698.55002
[24] Loops on the 3-sphere and umbral calculus, Algebraic topology, Evanston, IL (1988), 96, 297-302 (1989) · Zbl 0678.55002
[25] Universal constructions in umbral calculus, Mathematical essays in honor of Gian-Carlo Rota, Cambridge, MA (1996), 343-357 (1998) · Zbl 0908.05010
[26] Combinatorial identities (1979)
[27] The umbral calculus (1984) · Zbl 0536.33001
[28] A note on the Stong-Hattori theorem, Illinois J. Math., 17, 285-289 (1973) · Zbl 0267.55005
[29] Kummer congruences for the coefficients of Hurwitz series, Acta Arith., 40, 175-191 (1982) · Zbl 0482.10004
[30] Kummer congruences in formal groups and algebraic groups of dimension one, Rocky Mountain J. Math., 15, 1-11 (1985) · Zbl 0578.14041 · doi:10.1216/RMJ-1985-15-1-1
[31] Relations among characteristic numbers I, Topology, 4, 267-281 (1965) · Zbl 0136.20503 · doi:10.1016/0040-9383(65)90011-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.