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

Clarke, Francis; Hunton, John; Ray, Nigel

Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems
[Compléments au calcul "umbral" : les opérateurs "delta" doubles, les extensions de Leibniz et les théorèmes de Hattori-Stong]

Clarke, Francis ; Hunton, John ; Ray, Nigel

Annales de l'Institut Fourier, Tome 51 (2001), p. 297-336 / Harvested from Numdam

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Abstract

Nous poursuivons notre projet en vue d’étendre le calcul “umbral” de Roman-Rota au contexte des opérateurs “delta” sur un anneau gradué $E_{*}$ dans le but de développer des applications en topologie algébrique et en théorie des lois de groupes formels. Nous visons la situation où $E_{*}$ est libre de torsion additive; dans cette situation les questions centrales sont celles de la divisibilité. Nous étudions les algèbres de polynômes qui admettent l’action de deux opérateurs “delta” liés par une série inversible, et nous proposons des constructions connexes motivées par le théorème de Hattori-Stong en topologie algébrique. Notre traitement se poursuit exclusivement en termes de calcul “umbral”, ce qui nous mène à des applications topologiques nouvelles. En particulier, nous arrivons à une forme généralisée du théorème de Hattori-Stong.

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.

Publié le : 2001-01-01

MR 1824956 | Zbl 0962.05012

DOI : https://doi.org/10.5802/aif.1824
Classification: 05A40, 55N22
Mots clés: calcul umbral, théorèmes de Hattori-Stong

@article{AIF_2001__51_2_297_0,
     author = {Clarke, Francis and Hunton, John and Ray, Nigel},
     title = {Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems},
     journal = {Annales de l'Institut Fourier},
     volume = {51},
     year = {2001},
     pages = {297-336},
     doi = {10.5802/aif.1824},
     mrnumber = {1824956},
     zbl = {0962.05012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2001__51_2_297_0}
}

Clarke, Francis; Hunton, John; Ray, Nigel. Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems. Annales de l'Institut Fourier, Tome 51 (2001) pp. 297-336. doi : 10.5802/aif.1824. http://gdmltest.u-ga.fr/item/AIF_2001__51_2_297_0/

Bibliographie

[1] J. F. Adams On Chern characters and the structure of the unitary group, Proc. Cambridge Philos. Soc., Tome 57 (1961), pp. 189-199 | Article | MR 121795 | Zbl 0103.16001

[2] J. F. Adams Stable homotopy and generalised homology, University of Chicago Press, Chicago (1974) | MR 402720 | Zbl 0309.55016

[3] A. Baker Combinatorial and arithmetic identities based on formal group laws, Algebraic topology, Barcelona 1986, Springer, Berlin-New York (Lecture Notes in Math.) Tome 1298 (1987), pp. 17-34 | Zbl 0666.14019

[4] L. Carlitz Some properties of Hurwitz series, Duke Math. J, Tome 16 (1949), pp. 285-295 | Article | MR 29401 | Zbl 0041.17401

[5] F. Clarke The universal von Staudt theorems, Trans. Amer. Math. Soc., Tome 315 (1989), pp. 591-603 | Article | MR 986687 | Zbl 0683.10013

[6] L. Comtet Analyse Combinatoire, Presses Universitaires de France (1970) | MR 262087 | Zbl 0221.05001

[6] L. Comtet Advanced Combinatorics, D. Reidel Publishing Company, Dordrecht (1974) | MR 460128 | Zbl 0283.05001

[7] J. Dieudonné Sur les produits tensoriels, Ann. Sci. École Norm. Sup, Série 3, Tome 64 (1947), pp. 101-117 | Numdam | MR 23232 | Zbl 0033.24801

[8] A. Frohlich Formal groups, Springer, Berlin-Heidelberg, Lecture Notes in Math., Tome 74 (1968) | MR 242837 | Zbl 0177.04801

[9] L. Fuchs Abelian groups, Hungarian Acad. Sci., Budapest (1958) | MR 106942 | Zbl 0091.02704

[10] A. Hasse Die Gruppe der $p^{n}$ -primären Zahlen für einen Primteiler $𝔭$ von $p$ , J. Reine Angew. Math., Tome 176 (1936), pp. 174-183 | JFM 62.1115.01

[11] A. Hattori Integral characteristic numbers for weakly almost complex manifolds, Topology, Tome 5 (1966), pp. 259-280 | Article | MR 192517 | Zbl 0146.19401

[12] M. Hazewinkel Formal groups and applications, Academic Press, New York (1978) | MR 506881 | Zbl 0454.14020

[13] P. S. Landweber $B P_{*} (B P)$ and typical formal groups, Osaka J. Math., Tome 12 (1975), pp. 357-363 | MR 377945 | Zbl 0311.55003

[14] P. S. Landweber Homological properties of comodules over $M U_{*} (M U)$ and $B P_{*} (B P)$ , Amer. J. Math., Tome 98 (1976), pp. 591-610 | Article | MR 423332 | Zbl 0355.55007

[15] P. S. Landweber Supersingular elliptic curves and congruences for Legendre polynomials, Elliptic curves and modular forms in algebraic topology, Princeton (1986), Springer, Berlin-New York (Lecture Notes in Math.) Tome 1326 (1988), pp. 69-83 | Zbl 0655.14018

[16] G. Laures The topological $q$ -expansion principle, Topology, Tome 38 (1999), pp. 387-425 | Article | MR 1660325 | Zbl 0924.55004

[17] H. R. Miller; D. C. Ravenel Morava stabilizer algebras and the localisation of Novikov's $E_{2}$ -term, Duke Math. J., Tome 44 (1977), pp. 433-447 | Article | MR 458410 | Zbl 0358.55019

[18] J. Milnor On the cobordism ring $Ω^{*}$ and a complex analogue (part I), Amer. J. Math., Tome 82 (1960), pp. 505-521 | MR 119209 | Zbl 0095.16702

[19] D. Quillen On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc., Tome 75 (1969), pp. 1293-1298 | Article | MR 253350 | Zbl 0199.26705

[20] N. Ray Extensions of umbral calculus: penumbral coalgebras and generalised Bernoulli numbers, Adv. Math., Tome 61 (1986), pp. 49-100 | Article | MR 847728 | Zbl 0631.05002

[21] N. Ray Symbolic calculus: a 19th century approach to $M U$ and $B P$ , Homotopy theory, Durham (1985), Cambridge Univ. Press, Cambridge-New York (London Math. Soc. Lecture Note, Ser. 117) (1987), pp. 195-238 | Zbl 0651.55004

[22] N. Ray Stirling and Bernoulli numbers for complex oriented homology theories, Algebraic topology, Arcata, CA, (1986), Springer, Berlin-New York (Lecture Notes in Math.) Tome 1370 (1989), pp. 362-373 | Zbl 0698.55002

[23] N. Ray Loops on the 3-sphere and umbral calculus, Algebraic topology, Evanston, IL (1988), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 96 (1989), pp. 297-302 | Zbl 0678.55002

[24] N. Ray Universal constructions in umbral calculus, Mathematical essays in honor of Gian-Carlo Rota, Cambridge, MA (1996), Birkhäuser, Boston, MA (1998), pp. 343-357 | Zbl 0908.05010

[25] J. Riordan Combinatorial identities, Krieger, Huntington, NY (1979) | MR 554488

[26] S. Roman The umbral calculus, Academic Press, Orlando (1984) | MR 741185 | Zbl 0536.33001

[27] L. Smith A note on the Stong-Hattori theorem, Illinois J. Math., Tome 17 (1973), pp. 285-289 | MR 322852 | Zbl 0267.55005

[28] C. Snyder Kummer congruences for the coefficients of Hurwitz series, Acta Arith., Tome 40 (1982), pp. 175-191 | MR 649117 | Zbl 0482.10004

[29] C. Snyder Kummer congruences in formal groups and algebraic groups of dimension one, Rocky Mountain J. Math., Tome 15 (1985), pp. 1-11 | Article | MR 779247 | Zbl 0578.14041

[30] R. E. Stong Relations among characteristic numbers I, Topology, Tome 4 (1965), pp. 267-281 | Article | MR 192515 | Zbl 0136.20503