Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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SOME IDENTITIES RELATED TO THE EULER NUMBERS AND POLYNOMIALS

  • DOUK SOO JANG (Division of Mathematics, Science, and Computers, Kyungnam University)
  • Received : 2022.08.16
  • Accepted : 2022.12.22
  • Published : 2023.05.30
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Abstract

In this note, we give a proof of the p-adic analogue of mild generalization of classical zeta functions by modifying Osipov's method. In addition, we obtain some identities for the p-adic integration, from which, some classical formulas for Euler numbers and polynomials have been deduced.

Keywords

Acknowledgement

The author would like to thank the referees for their valuable comments and suggestions which improved the presentation of this paper.

References

  1. A. Adelberg, Congruences of p-adic integer order Bernoulli numbers, J. Number Theory 59 (1996), 374-388. https://doi.org/10.1006/jnth.1996.0103
  2. E.W. Barnes, On the theory of the multiple Gamma function, Trans. Cambridge Philos. Soc. 19 (1904), 374-425.
  3. L. Carlitz, Some theorems on Bernoulli numbers of higher order, Pacific J. Math. 2 (1952), 127-139. https://doi.org/10.2140/pjm.1952.2.127
  4. L. Carlitz, Some congruences for the Bernoulli numbers, Amer. J. Math. 75 (1953), 163-172. https://doi.org/10.2307/2372625
  5. L. Carlitz and F.R. Olson, Some theorems on Bernoulli and Euler numbers of higher order, Duke Math. J. 21 (1954), 405-421.
  6. J. Diamond, The p-adic log gamma function and p-adic Euler constants, Trans. Amer. Math. Soc. 233 (1977), 321-337.
  7. G. Frobenius, Uber die Bernoullischen Zahlen un die Eulerschen Polynome, Sitz. Preuss. Akad. Wiss. (1910), 809-847.
  8. F.T. Howard, Congruences and recurrences for Bernoulli numbers of higher order, Fibonacci Quart. 32 (1994), 316-328.
  9. K. Iwasawa, Lectures on p-adic L-functions, Annals of Mathematics Studies, No. 74, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972.
  10. A. Yu. Khrennikov and M. Nilsson, p-adic deterministic and random dynamics, Mathematics and its Applications, 574, Kluwer Academic Publishers, Dordrecht, 2004.
  11. M.-S. Kim and S.-W. Son, Bernoulli numbers in p-adic analysis, Appl. Math. Comput. 39 (2002), 221-236. https://doi.org/10.4134/JKMS.2002.39.2.221
  12. T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), 261-267.
  13. N. Koblitz, A new proof of certain formulas for p-adic L-functions, Duke Math. J. 46 (1979), 455-468. https://doi.org/10.1215/S0012-7094-79-04621-0
  14. N. Koblitz, p-Adic Analysis: a Short Course on Recent Work, Cambridge University Press, Mathematical Society Lecture Notes Series 46, 1980.
  15. T. Kubota und H.W. Leopoldt, Eine p-adische Theorie der Zetawerte. I. Einfuhrung der p-adischen Dirichletschen L-Funktionen, J. Reine Angew. Math. 214/215 (1964), 328-339. https://doi.org/10.1515/crll.1964.214-215.328
  16. E.E. Kummer, Uber eine allgemeine Eigenschaft der rationalen Entwickelungscoeficienten einer bestimmten Gattung analytischer Funktionen, J. Reine Angew. Math. 41 (1851), 368-372. https://doi.org/10.1515/crll.1851.41.368
  17. T. Momotani, Regularized product expressions of higher Riemann zeta functions, Proc. Amer. Math. Soc. 134 (2006), 2541-2548. https://doi.org/10.1090/S0002-9939-06-08291-8
  18. Yu.V. Osipov, p-adic zeta functions and Bernoulli numbers, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 93 (1980), 192-203; English transl. in Journal of Mathematical Sciences 19 (1982), 1186-1194. https://doi.org/10.1007/BF01085132
  19. K. Ota, On Kummer-type congruences for derivatives of Barnes' multiple Bernoulli polynomials, J. Number Theory 92 (2002), 1-36. https://doi.org/10.1006/jnth.2001.2702
  20. D. Park and J.-W. Son, On a p-adic analogue of k-ple Riemann zeta function, Bull. Korean Math. Soc. 49 (2012), 165-174. https://doi.org/10.4134/BKMS.2012.49.1.165
  21. H.S. Vandiver, Certain congruences involving the Bernoulli numbers, Duke Math. J. 5 (1939), 548-551. https://doi.org/10.1215/S0012-7094-39-00546-6
  22. L.C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Graduate Texts in Mathematics 83, Springer-Verlag, New York, 1997.
  23. P.T. Young, Congruences for Bernoulli, Euler, and Stirling numbers, J. Number Theory 78 (1999), 204-227. https://doi.org/10.1006/jnth.1999.2401