Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The universal von Staudt theorems


Author: Francis Clarke
Journal: Trans. Amer. Math. Soc. 315 (1989), 591-603
MSC: Primary 11B68; Secondary 05A19, 12E10, 33A25
MathSciNet review: 986687
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove general forms of von Staudt's theorems on the Bernoulli numbers. As a consequence we are able to deduce strong versions of a number of congruences involving various generalisations of the Bernoulli numbers. For example we obtain an improved form of a congruence due to Hurwitz involving the Laurent series coefficients of the Weierstrass elliptic function associated with a square lattice.


References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, On the groups 𝐽(𝑋). II, Topology 3 (1965), 137–171. MR 0198468 (33 #6626)
  • [2] Andrew Baker, Combinatorial and arithmetic identities based on formal group laws, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 17–34. MR 928821 (89f:55012), http://dx.doi.org/10.1007/BFb0082998
  • [3] L. Carlitz, The coefficients of the reciprocal of a series, Duke Math. J. 8 (1941), 689–700. MR 0005392 (3,147j)
  • [4] L. Carlitz, Some congruences for the Bernoulli numbers, Amer. J. Math. 75 (1953), 163–172. MR 0051871 (14,539c)
  • [5] L. Carlitz, A note on the Staudt-Clausen theorem, Amer. Math. Monthly 64 (1957), 19–21. MR 0082497 (18,560c)
  • [6] L. Carlitz, Classroom Notes: A Property of the Bernoulli Numbers, Amer. Math. Monthly 66 (1959), no. 8, 714–715. MR 1530459, http://dx.doi.org/10.2307/2309351
  • [7] T. Clausen, Theorem, Astronomische Nachrichten 17 (1840), 351-352.
  • [8] Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128 (57 #124)
  • [9] I. Dibag, An analogue of the von Staudt-Clausen theorem, J. Algebra 87 (1984), no. 2, 332–341. MR 739937 (85j:11028), http://dx.doi.org/10.1016/0021-8693(84)90140-6
  • [10] G. Frobenius, Über die Bernoullischen Zahlen und die Eulerschen Polynome, Sitzungsberichte Berliner Akademie der Wissenschaften, 1910, pp. 809-847.
  • [11] C. W. Haigh, Newton 's identities, generalised cycle-indices, universal Bernoulli numbers and truncated Schur-functions, J. Math. Chem. (submitted).
  • [12] A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann. 51 (1899), 196-226.
  • [13] Nicholas M. Katz, The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers, Math. Ann. 216 (1975), 1–4. MR 0387293 (52 #8136)
  • [14] Haynes Miller, Universal Bernoulli numbers and the 𝑆¹-transfer, Current trends in algebraic topology, Part 2 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 1982, pp. 437–449. MR 686158 (85b:55029)
  • [15] John W. Milnor and James D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76. MR 0440554 (55 #13428)
  • [16] N. Nielsen, Traité élémentaire des nombres de Bernoulli, Gauthier-Villars, Paris, 1923.
  • [17] Nigel Ray, Extensions of umbral calculus: penumbral coalgebras and generalised Bernoulli numbers, Adv. in Math. 61 (1986), no. 1, 49–100. MR 847728 (88b:05019), http://dx.doi.org/10.1016/0001-8708(86)90065-4
  • [18] -, Symbolic calculus: a 19th century approach to $ MU$ and $ BP$, Homotopy Theory (E. Rees and J. D. S. Jones, eds.), Proceedings of the Durham Symposium, 1985, London Math. Soc. Lecture Note Ser. No. 117, Cambridge Univ. Press, Cambridge, 1987, pp. 195-238.
  • [19] -, Stirling and Bernoulli numbers for complex oriented homology theories, Proc. Internat. Conf. Algebraic Topology, 1986, Arcata (G. Carlsson, R. L. Cohen, H. R. Miller and D. C. Ravenel, eds.), Lecture Notes in Math. vol. 1370, Springer-Verlag, Berlin and New York, 1989, pp. 362-363.
  • [20] Chip Snyder, A concept of Bernoulli numbers in algebraic function fields, J. Reine Angew. Math. 307/308 (1979), 295–308. MR 534227 (81a:12016), http://dx.doi.org/10.1515/crll.1979.307-308.295
  • [21] K. G. C. von Staudt, Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffend, J. Reine Angew. Math. 21 (1840), 373-374.
  • [22] -, De numeris Bernoullianis, Erlangen, 1845.
  • [23] -, De numeris Bernoullianis, commentatio altera, Erlangen, 1845.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11B68, 05A19, 12E10, 33A25

Retrieve articles in all journals with MSC: 11B68, 05A19, 12E10, 33A25


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0986687-3
PII: S 0002-9947(1989)0986687-3
Keywords: Bernoulli numbers
Article copyright: © Copyright 1989 American Mathematical Society