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

Clarke, Francis; Hunton, John; Ray, Nigel

Annales de l'Institut Fourier, Tome 51 (2001) no. 2, pp. 297-336.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

[13] P. S. Landweber $B{P}_{*}\left(BP\right)$ and typical formal groups, Osaka J. Math., Volume 12 (1975), pp. 357-363 | MR | Zbl

[14] P. S. Landweber Homological properties of comodules over $M{U}_{*}\left(MU\right)$ and $B{P}_{*}\left(BP\right)$, Amer. J. Math., Volume 98 (1976), pp. 591-610 | DOI | MR | Zbl

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

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

[17] H. R. Miller; D. C. Ravenel Morava stabilizer algebras and the localisation of Novikov's $E_2$-term, Duke Math. J., Volume 44 (1977), pp. 433-447 | DOI | MR | Zbl

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

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

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

[21] N. Ray Symbolic calculus: a 19th century approach to $MU$ and $BP$, Homotopy theory, Durham (1985) (London Math. Soc. Lecture Note, Ser. 117) (1987), pp. 195-238 | Zbl

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

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

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

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

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

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

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

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

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