대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제56권2호
  • /
  • Pages.491-499
  • /
  • 2019
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ON LACUNARY RECURRENCES WITH GAPS OF LENGTH FOUR AND EIGHT FOR THE BERNOULLI NUMBERS

  • 투고 : 2018.04.10
  • 심사 : 2018.08.22
  • 발행 : 2019.03.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

The problem of finding fast computing methods for Bernoulli numbers has a long and interesting history. In this paper, the author provides new proofs for two lacunary recurrence relations with gaps of length four and eight for the Bernoulli numbers. These proofs invoked the fact that the nth powers of ${\pi}^2$, ${\pi}^4$ and ${\pi}^8$ can be expressed in terms of the nth elementary symmetric functions.

키워드

참고문헌

  1. J. G. F. Belinfante and I. Gessel, Problems and Solutions: Solutions of Elementary Problems: E3237, Amer. Math. Monthly 96 (1989), no. 4, 364-365. https://doi.org/10.2307/2324102
  2. B. C. Berndt, Ramanujan's notebooks. Part IV, Springer-Verlag, New York, 1994.
  3. M. Chellali, Acceleration de calcul de nombres de Bernoulli, J. Number Theory 28 (1988), no. 3, 347-362. https://doi.org/10.1016/0022-314X(88)90047-9
  4. R. Honsberger, Mathematical Gems. III, The Dolciani Mathematical Expositions, 9, Mathematical Association of America, Washington, DC, 1985.
  5. F. T. Howard, Applications of a recurrence for the Bernoulli numbers, J. Number Theory 52 (1995), no. 1, 157-172. https://doi.org/10.1006/jnth.1995.1062
  6. F. T. Howard, Formulas of Ramanujan involving Lucas numbers, Pell numbers, and Bernoulli numbers, in Applications of Fibonacci numbers, Vol. 6 (Pullman, WA, 1994), 257-270, Kluwer Acad. Publ., Dordrecht, 1996.
  7. F. T. Howard, A general lacunary recurrence formula, in Applications of Fibonacci numbers. Vol. 9, 121-135, Kluwer Acad. Publ., Dordrecht, 2004.
  8. K. F. Ireland and M. I. Rosen, A Classical Introduction to Modern Number Theory, revised edition of Elements of number theory, Graduate Texts in Mathematics, 84, Springer-Verlag, New York, 1982.
  9. D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Ann. of Math. (2) 36 (1935), no. 3, 637-649. https://doi.org/10.2307/1968647
  10. M. Merca, Asymptotics of the Chebyshev-Stirling numbers of the first kind, Integral Transforms Spec. Funct. 27 (2016), no. 4, 259-267. https://doi.org/10.1080/10652469.2015.1117460
  11. M. Merca, On families of linear recurrence relations for the special values of the Riemann zeta function, J. Number Theory 170 (2017), 55-65. https://doi.org/10.1016/j.jnt.2016.06.015
  12. M. Merca, Two algorithms for computing the Riemann zeta functions ${\zeta}$(4n) and ${\zeta}$(8n) as a reduced fraction, submitted to publication.
  13. V. Namias, A simple derivation of Stirling's asymptotic series, Amer. Math. Monthly 93 (1986), no. 1, 25-29. https://doi.org/10.1080/00029890.1986.11971738
  14. S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc. 3 (1911), 219-234.
  15. J. Riordan, Combinatorial Identities, John Wiley & Sons, Inc., New York, 1968.
  16. T. Simpson, The invention of a general method for determining the sum of every 2d, 3d, 4th, or 5th, & c, term of a series, taken in order; the sum of the whole series being known, Philosophical Transactions 50 (1757-1758), 757-769, available at http://www.jstor.org/stable/105328.
  17. H. Tsumura, An elementary proof of Euler's formula for ${\zeta}$(2m), Amer. Math. Monthly 111 (2004), no. 5, 430-431. https://doi.org/10.2307/4145270
  18. C. C. Yalavigi, Bernoulli and Lucas numbers, Math. Education 5 (1971), A99-A102.