Year 2013,
Volume: 42 Issue: 4
,
319
-
329
,
01.04.2013
Abstract
Inspired by a result of Saalsch¨utz, we prove a recurrence relation forBernoulli numbers. This recurrence relation has an interesting connection with real cyclotomic fields.
References
- Agoh, T. and Dilcher, K. Shortened recurrence relations for Bernoulli numbers, Discrete Math. 309(4), 887–898, 2009.
- Apostol, T. M. An elementary view of Euler’s summation formula, Amer. Math. Monthly 106 (5), 409–418, 1999.
- Bromwich, T. J. I’A. An Introduction to the Theory of Infinite Series. Second edition. London, Macmillan; New York, St. Martin’s Press, 1965.
- Cox, D., Little, J. and O’Shea, D. Ideals, varieties, and algorithms. Second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1997.
- Gould, H. W. Explicit formulas for Bernoulli numbers, Amer. Math. Monthly 79, 44–51, 197 Harvey, D. A multimodular algorithm for computing Bernoulli numbers, Math. Comp. 79 (272), 2361–2370, 2010.
- Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory. Second edition. Graduate Texts in Mathematics, 84. Springer-Verlag, New York, 1990.
- Kummer, E. E. Allgemeiner Beweis des Fermat’schen Satzes, dass die Gleichung x λ + y λ = z λ durch ganze Zahlenunl¨ osbar ist, f¨ ur alle diejenigen Potenz-Exponenten λ, welche ungerade Primzahlen sind und in den Z¨ ahlern der ersten (λ − 3)/2 Bernoulli’schen Zahlen als Factoren nicht vorkommen, J. Reine Angew. Math. 40, 131–138, 1850.
- Mills, S. The independent derivations by Leonhard Euler and Colin Maclaurin of the EulerMaclaurin summation formula, Arch. Hist. Exact Sci. 33 (1-3), 1–13, 1985.
- Saalsch¨ utz, L. Neue Formeln f¨ ur die Bernoullischen Zahlen, J. Reine Angew. Math. 126, 99–101, 1903.
- Sloane, N.J.A. On-Line Encyclopedia of Integer Sequences, http://www.oeis.org. Washington, L. C. Introduction to cyclotomic fields. Second edition. Graduate Texts in Mathematics, 83. Springer-Verlag, New York, 1997.
Year 2013,
Volume: 42 Issue: 4
,
319
-
329
,
01.04.2013
Abstract
-
Keywords
References
- Agoh, T. and Dilcher, K. Shortened recurrence relations for Bernoulli numbers, Discrete Math. 309(4), 887–898, 2009.
- Apostol, T. M. An elementary view of Euler’s summation formula, Amer. Math. Monthly 106 (5), 409–418, 1999.
- Bromwich, T. J. I’A. An Introduction to the Theory of Infinite Series. Second edition. London, Macmillan; New York, St. Martin’s Press, 1965.
- Cox, D., Little, J. and O’Shea, D. Ideals, varieties, and algorithms. Second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1997.
- Gould, H. W. Explicit formulas for Bernoulli numbers, Amer. Math. Monthly 79, 44–51, 197 Harvey, D. A multimodular algorithm for computing Bernoulli numbers, Math. Comp. 79 (272), 2361–2370, 2010.
- Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory. Second edition. Graduate Texts in Mathematics, 84. Springer-Verlag, New York, 1990.
- Kummer, E. E. Allgemeiner Beweis des Fermat’schen Satzes, dass die Gleichung x λ + y λ = z λ durch ganze Zahlenunl¨ osbar ist, f¨ ur alle diejenigen Potenz-Exponenten λ, welche ungerade Primzahlen sind und in den Z¨ ahlern der ersten (λ − 3)/2 Bernoulli’schen Zahlen als Factoren nicht vorkommen, J. Reine Angew. Math. 40, 131–138, 1850.
- Mills, S. The independent derivations by Leonhard Euler and Colin Maclaurin of the EulerMaclaurin summation formula, Arch. Hist. Exact Sci. 33 (1-3), 1–13, 1985.
- Saalsch¨ utz, L. Neue Formeln f¨ ur die Bernoullischen Zahlen, J. Reine Angew. Math. 126, 99–101, 1903.
- Sloane, N.J.A. On-Line Encyclopedia of Integer Sequences, http://www.oeis.org. Washington, L. C. Introduction to cyclotomic fields. Second edition. Graduate Texts in Mathematics, 83. Springer-Verlag, New York, 1997.
There are 10 citations in total.
Details
| Primary Language | Turkish |
|---|---|
| Authors | |
| Publication Date | April 1, 2013 |
| IZ | https://izlik.org/JA79UH33XZ |
| Published in Issue | Year 2013 Volume: 42 Issue: 4 |