Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T15:05:57.627Z Has data issue: false hasContentIssue false
  • English
  • Français

On a multidimensional Volkenborn integral and higher order Bernoulli numbers

Published online by Cambridge University Press:  17 April 2009

Min-Soo Kim  and
Jin-Woo Son
Min-Soo Kim
Affiliation:
Department of Mathematics, Kyungnam University, Masan 631–701, South Korea e-mail: [email protected]
Jin-Woo Son
Affiliation:
Department of Mathematics, Kyungnam University, Masan 631–701, South Korea e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, using a multidimensional Volkenborn integral, we give a p-adic expression of the higher order Bernoulli numbers. This shows immediately the relation to the sums of products of the ordinary Bernoulli numbers of Dilcher in 1996. We also consider the Mahler expansion of several p-adic variables function, and give some examples.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 1 , February 2002 , pp. 59 - 71
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Adelberg, A., ‘Congruences of p-adic integer order Bernoulli numbers’, J. Number Theory 59 (1996), 374–388.Google Scholar
[2]Carlitz, L., ‘Some theorems on Bernoulli numbers of higher order’, Pacific J. Math. 2 (1952), 127–139.Google Scholar
[3]Diamond, J., ‘The p-adic log gamma function and p-adic Euler constants’, Trans. Amer. Math. Soc. 233 (1977), 321–337.Google Scholar
[4]Dilcher, K., ‘Sums of products of Bernoulli numbers’, J. Number Theory 60 (1996), 23–41.CrossRefGoogle Scholar
[5]Howard, F. T., ‘Congruences and recurrences for Bernoulli numbers of higher order’, Fibonacci Quart. 32 (1994), 316–328.Google Scholar
[6]Koblitz, N., p-Adic Analysis: a Short Course on Recent Work, London Mathematical Society Lecture Notes 46 (Cambridge University Press, Cambridge, New York, 1980).Google Scholar
[7]Koblitz, N., p-adic Numbers, p-adic Analysis and Zeta-Functions, Graduate Texts in Mathematics 58 (Springer-Verlag, Berlin, Heidelberg, New York, 1984).CrossRefGoogle Scholar
[8]Mahler, K., p-adic numbers and their functions, Cambridge tracts in mathematics 76 (Cambridge University Press, Cambridge, New York, 1981).Google Scholar
[9]Nörlund, N. E., Vorlesungen über Differenzenrechnung (Springer, Berlin, 1924).CrossRefGoogle Scholar
[10]Robert, A. M.. A Course in p-adic Analysis (Springer-Verlag, Berlin, Heidelberg, New York, 2000).Google Scholar
[11]Schikhof, W. H., Ultrametric Calculus, An introduction to p-adic analysis, Cambridge Studies in Adv. Math. 4 (Cambridge University Press, Cambridge, New York, 1984).Google Scholar
[12]Washington, L. C., Introduction to Cyclotomic Fields, Graduate Texts in Mathematics 83 (Springer-Verlag, Berlin, Heidelberg, New York, 1997).Google Scholar
[13]Woodcock, C. F., ‘An invariant p-adic integral on ℤp’, J. London Math. Soc. 8 (1974), 731–734.Google Scholar
[14]Young, P. T., ‘Congruences for Bernoulli, Euler, and Stirling numbers’, J. Number Theory 78 (1999), 204–227.Google Scholar