Bernoulli number (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Bernoulli number" in English language version.

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Stanley, Richard P. (2010), "A survey of alternating permutations", Combinatorics and graphs, Contemporary Mathematics, vol. 531, Providence, RI: American Mathematical Society, pp. 165–196, arXiv:0912.4240, doi:10.1090/conm/531/10466, ISBN 978-0-8218-4865-4, MR 2757798, S2CID 14619581
Neukirch, Jürgen (1999), Algebraische Zahlentheorie, Grundlehren der mathematischen Wissenschaften, vol. 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021 §VII.2.

archive.org

Translation of the text: " ... And if [one were] to proceed onward step by step to higher powers, one may furnish, with little difficulty, the following list:
Sums of powers
$\textstyle \int n=\sum _{k=1}^{n}k={\frac {1}{2}}n^{2}+{\frac {1}{2}}n$ $\textstyle \int n=\sum _{k=1}^{n}k={\frac {1}{2}}n^{2}+{\frac {1}{2}}n$

⋮
$\textstyle \int n^{10}=\sum _{k=1}^{n}k^{10}={\frac {1}{11}}n^{11}+{\frac {1}{2}}n^{10}+{\frac {5}{6}}n^{9}-1n^{7}+1n^{5}-{\frac {1}{2}}n^{3}+{\frac {5}{66}}n$ $\textstyle \int n^{10}=\sum _{k=1}^{n}k^{10}={\frac {1}{11}}n^{11}+{\frac {1}{2}}n^{10}+{\frac {5}{6}}n^{9}-1n^{7}+1n^{5}-{\frac {1}{2}}n^{3}+{\frac {5}{66}}n$
Indeed [if] one will have examined diligently the law of arithmetic progression there, one will also be able to continue the same without these circuitous computations: For [if] $\textstyle c$ $\textstyle c$ is taken as the exponent of any power, the sum of all $\textstyle n^{c}$ $\textstyle n^{c}$ is produced or
$\textstyle \int n^{c}=\sum _{k=1}^{n}k^{c}={\frac {1}{c+1}}n^{c+1}+{\frac {1}{2}}n^{c}+{\frac {c}{2}}An^{c-1}+{\frac {c(c-1)(c-2)}{2\cdot 3\cdot 4}}Bn^{c-3}+{\frac {c(c-1)(c-2)(c-3)(c-4)}{2\cdot 3\cdot 4\cdot 5\cdot 6}}Cn^{c-5}+{\frac {c(c-1)(c-2)(c-3)(c-4)(c-5)(c-6)}{2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8}}Dn^{c-7}+\cdots$ $\textstyle \int n^{c}=\sum _{k=1}^{n}k^{c}={\frac {1}{c+1}}n^{c+1}+{\frac {1}{2}}n^{c}+{\frac {c}{2}}An^{c-1}+{\frac {c(c-1)(c-2)}{2\cdot 3\cdot 4}}Bn^{c-3}+{\frac {c(c-1)(c-2)(c-3)(c-4)}{2\cdot 3\cdot 4\cdot 5\cdot 6}}Cn^{c-5}+{\frac {c(c-1)(c-2)(c-3)(c-4)(c-5)(c-6)}{2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8}}Dn^{c-7}+\cdots$
and so forth, the exponent of its power $n$ $n$ continually diminishing by 2 until it arrives at $n$ $n$ or $n^{2}$ $n^{2}$ . The capital letters $\textstyle A,B,C,D,$ $\textstyle A,B,C,D,$ etc. denote in order the coefficients of the last terms for $\textstyle \int n^{2},\int n^{4},\int n^{6},\int n^{8}$ $\textstyle \int n^{2},\int n^{4},\int n^{6},\int n^{8}$ , etc. namely
$\textstyle A={\frac {1}{6}},B=-{\frac {1}{30}},C={\frac {1}{42}},D=-{\frac {1}{30}}$ $\textstyle A={\frac {1}{6}},B=-{\frac {1}{30}},C={\frac {1}{42}},D=-{\frac {1}{30}}$ ."
[Note: The text of the illustration contains some typos: ensperexit should read inspexerit, ambabimus should read ambagibus, quosque should read quousque, and in Bernoulli's original text Sumtâ should read Sumptâ or Sumptam.]
- Smith, David Eugene (1929), "Jacques (I) Bernoulli: On the 'Bernoulli Numbers'", A Source Book in Mathematics, New York: McGraw-Hill Book Co., pp. 85–90

arxiv.org

Knuth (1993). Knuth, D. E. (1993), "Johann Faulhaber and the Sums of Powers", Mathematics of Computation, 61 (203), American Mathematical Society: 277–294, arXiv:math/9207222, doi:10.2307/2152953, JSTOR 2152953
Knuth (1993), p. 14. Knuth, D. E. (1993), "Johann Faulhaber and the Sums of Powers", Mathematics of Computation, 61 (203), American Mathematical Society: 277–294, arXiv:math/9207222, doi:10.2307/2152953, JSTOR 2152953
Harvey, David (2010), "A multimodular algorithm for computing Bernoulli numbers", Math. Comput., 79 (272): 2361–2370, arXiv:0807.1347, doi:10.1090/S0025-5718-2010-02367-1, S2CID 11329343, Zbl 1215.11016
Guo, Victor J. W.; Zeng, Jiang (30 August 2005), "A q-Analogue of Faulhaber's Formula for Sums of Powers", The Electronic Journal of Combinatorics, 11 (2), arXiv:math/0501441, Bibcode:2005math......1441G, doi:10.37236/1876, S2CID 10467873
Stanley, Richard P. (2010), "A survey of alternating permutations", Combinatorics and graphs, Contemporary Mathematics, vol. 531, Providence, RI: American Mathematical Society, pp. 165–196, arXiv:0912.4240, doi:10.1090/conm/531/10466, ISBN 978-0-8218-4865-4, MR 2757798, S2CID 14619581
Elkies, N. D. (2003), "On the sums Sum_(k=-infinity...infinity) (4k+1)^(-n)", Amer. Math. Monthly, 110 (7): 561–573, arXiv:math.CA/0101168, doi:10.2307/3647742, JSTOR 3647742
Euler, Leonhard (1735), "De summis serierum reciprocarum", Opera Omnia, I.14, E 41: 73–86, arXiv:math/0506415, Bibcode:2005math......6415E
Bannai, Kenichi; Kobayashi, Shinichi (2010), "Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers", Duke Mathematical Journal, 153 (2), arXiv:math/0610163, doi:10.1215/00127094-2010-024, ISSN 0012-7094, S2CID 9262012
Malenfant, Jerome (2011), "Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers", arXiv:1103.1585 [math.NT]

bernoulli.org

Kellner, Bernd (2002), Program Calcbn – A program for calculating Bernoulli numbers.

books.google.com

Smith, David Eugene; Mikami, Yoshio (1914), A history of Japanese mathematics, Open Court publishing company, p. 108, ISBN 9780486434827

cornell.edu

digital.library.cornell.edu

Saalschütz, Louis (1893), Vorlesungen über die Bernoullischen Zahlen, ihren Zusammenhang mit den Secanten-Coefficienten und ihre wichtigeren Anwendungen, Berlin: Julius Springer.

digizeitschriften.de

Kummer, E. E. (1850), "Allgemeiner Beweis des Fermat'schen Satzes, dass die Gleichung x^λ + y^λ = z^λ durch ganze Zahlen unlösbar ist, für alle diejenigen Potenz-Exponenten λ, welche ungerade Primzahlen sind und in den Zählern der ersten (λ-3)/2 Bernoulli'schen Zahlen als Factoren nicht vorkommen", J. Reine Angew. Math., 40: 131–138

doi.org

Kitagawa, Tomoko L. (2021-07-23), "The Origin of the Bernoulli Numbers: Mathematics in Basel and Edo in the Early Eighteenth Century", The Mathematical Intelligencer, 44: 46–56, doi:10.1007/s00283-021-10072-y, ISSN 0343-6993
Knuth (1993). Knuth, D. E. (1993), "Johann Faulhaber and the Sums of Powers", Mathematics of Computation, 61 (203), American Mathematical Society: 277–294, arXiv:math/9207222, doi:10.2307/2152953, JSTOR 2152953
Knuth (1993), p. 14. Knuth, D. E. (1993), "Johann Faulhaber and the Sums of Powers", Mathematics of Computation, 61 (203), American Mathematical Society: 277–294, arXiv:math/9207222, doi:10.2307/2152953, JSTOR 2152953
Buhler, J.; Crandall, R.; Ernvall, R.; Metsankyla, T.; Shokrollahi, M. (2001), "Irregular Primes and Cyclotomic Invariants to 12 Million", Journal of Symbolic Computation, 31 (1–2): 89–96, doi:10.1006/jsco.1999.1011
Harvey, David (2010), "A multimodular algorithm for computing Bernoulli numbers", Math. Comput., 79 (272): 2361–2370, arXiv:0807.1347, doi:10.1090/S0025-5718-2010-02367-1, S2CID 11329343, Zbl 1215.11016
Guo, Victor J. W.; Zeng, Jiang (30 August 2005), "A q-Analogue of Faulhaber's Formula for Sums of Powers", The Electronic Journal of Combinatorics, 11 (2), arXiv:math/0501441, Bibcode:2005math......1441G, doi:10.37236/1876, S2CID 10467873
Gould, Henry W. (1972), "Explicit formulas for Bernoulli numbers", Amer. Math. Monthly, 79 (1): 44–51, doi:10.2307/2978125, JSTOR 2978125
Woon, S. C. (1997), "A tree for generating Bernoulli numbers", Math. Mag., 70 (1): 51–56, doi:10.2307/2691054, JSTOR 2691054
Stanley, Richard P. (2010), "A survey of alternating permutations", Combinatorics and graphs, Contemporary Mathematics, vol. 531, Providence, RI: American Mathematical Society, pp. 165–196, arXiv:0912.4240, doi:10.1090/conm/531/10466, ISBN 978-0-8218-4865-4, MR 2757798, S2CID 14619581
Elkies, N. D. (2003), "On the sums Sum_(k=-infinity...infinity) (4k+1)^(-n)", Amer. Math. Monthly, 110 (7): 561–573, arXiv:math.CA/0101168, doi:10.2307/3647742, JSTOR 3647742
Knuth, D. E.; Buckholtz, T. J. (1967), "Computation of Tangent, Euler, and Bernoulli Numbers", Mathematics of Computation, 21 (100), American Mathematical Society: 663–688, doi:10.2307/2005010, JSTOR 2005010
Arnold, V. I. (1991), "Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics", Duke Math. J., 63 (2): 537–555, doi:10.1215/s0012-7094-91-06323-4
Clausen, Thomas (1840), "Lehrsatz aus einer Abhandlung über die Bernoullischen Zahlen", Astron. Nachr., 17 (22): 351–352, doi:10.1002/asna.18400172205
Riesz, M. (1916), "Sur l'hypothèse de Riemann", Acta Mathematica, 40: 185–90, doi:10.1007/BF02418544
Charollois, Pierre; Sczech, Robert (2016), "Elliptic Functions According to Eisenstein and Kronecker: An Update", EMS Newsletter, 2016–9 (101): 8–14, doi:10.4171/NEWS/101/4, ISSN 1027-488X, S2CID 54504376
Bannai, Kenichi; Kobayashi, Shinichi (2010), "Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers", Duke Mathematical Journal, 153 (2), arXiv:math/0610163, doi:10.1215/00127094-2010-024, ISSN 0012-7094, S2CID 9262012
Carlitz, L. (1968), "Bernoulli Numbers", Fibonacci Quarterly, 6 (3): 71–85, doi:10.1080/00150517.1968.12431229
Agoh, Takashi; Dilcher, Karl (2008), "Reciprocity Relations for Bernoulli Numbers", American Mathematical Monthly, 115 (3): 237–244, doi:10.1080/00029890.2008.11920520, JSTOR 27642447, S2CID 43614118

eudml.org

Kummer, E. E. (1851), "Über eine allgemeine Eigenschaft der rationalen Entwicklungscoefficienten einer bestimmten Gattung analytischer Functionen", J. Reine Angew. Math., 1851 (41): 368–372

fourmilab.ch

Menabrea, L.F. (1842), "Sketch of the Analytic Engine invented by Charles Babbage, with notes upon the Memoir by the Translator Ada Augusta, Countess of Lovelace", Bibliothèque Universelle de Genève, 82, See Note G

harvard.edu

ui.adsabs.harvard.edu

Selin, Helaine, ed. (1997), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, p. 819 (p. 891), Bibcode:2008ehst.book.....S, ISBN 0-7923-4066-3
Guo, Victor J. W.; Zeng, Jiang (30 August 2005), "A q-Analogue of Faulhaber's Formula for Sums of Powers", The Electronic Journal of Combinatorics, 11 (2), arXiv:math/0501441, Bibcode:2005math......1441G, doi:10.37236/1876, S2CID 10467873
Euler, Leonhard (1735), "De summis serierum reciprocarum", Opera Omnia, I.14, E 41: 73–86, arXiv:math/0506415, Bibcode:2005math......6415E

jstor.org

Knuth (1993). Knuth, D. E. (1993), "Johann Faulhaber and the Sums of Powers", Mathematics of Computation, 61 (203), American Mathematical Society: 277–294, arXiv:math/9207222, doi:10.2307/2152953, JSTOR 2152953
Knuth (1993), p. 14. Knuth, D. E. (1993), "Johann Faulhaber and the Sums of Powers", Mathematics of Computation, 61 (203), American Mathematical Society: 277–294, arXiv:math/9207222, doi:10.2307/2152953, JSTOR 2152953
Gould, Henry W. (1972), "Explicit formulas for Bernoulli numbers", Amer. Math. Monthly, 79 (1): 44–51, doi:10.2307/2978125, JSTOR 2978125
Woon, S. C. (1997), "A tree for generating Bernoulli numbers", Math. Mag., 70 (1): 51–56, doi:10.2307/2691054, JSTOR 2691054
Elkies, N. D. (2003), "On the sums Sum_(k=-infinity...infinity) (4k+1)^(-n)", Amer. Math. Monthly, 110 (7): 561–573, arXiv:math.CA/0101168, doi:10.2307/3647742, JSTOR 3647742
Knuth, D. E.; Buckholtz, T. J. (1967), "Computation of Tangent, Euler, and Bernoulli Numbers", Mathematics of Computation, 21 (100), American Mathematical Society: 663–688, doi:10.2307/2005010, JSTOR 2005010
Agoh, Takashi; Dilcher, Karl (2008), "Reciprocity Relations for Bernoulli Numbers", American Mathematical Monthly, 115 (3): 237–244, doi:10.1080/00029890.2008.11920520, JSTOR 27642447, S2CID 43614118

luschny.de

Peter Luschny (2013), The Bernoulli Manifesto

maecla.it

this formula was discovered (or perhaps rediscovered) by Giorgio Pietrocola. His demonstration is available in Italian language (Pietrocola 2008). Pietrocola, Giorgio (October 31, 2008), "Esplorando un antico sentiero: teoremi sulla somma di potenze di interi successivi (Corollario 2b)", Maecla (in Italian), retrieved April 8, 2017.

nodak.edu

genealogy.math.ndsu.nodak.edu

The Mathematics Genealogy Project (n.d.) shows Leibniz as the academic advisor of Jakob Bernoulli. See also Miller (2017). The Mathematics Genealogy Project, Fargo: Department of Mathematics, North Dakota State University, n.d., archived from the original on 10 May 2019, retrieved 11 May 2019. Miller, Jeff (23 June 2017), "Earliest Uses of Symbols of Calculus", Earliest Uses of Various Mathematical Symbols, retrieved 11 May 2019.

projecteuclid.org

Bannai, Kenichi; Kobayashi, Shinichi (2010), "Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers", Duke Mathematical Journal, 153 (2), arXiv:math/0610163, doi:10.1215/00127094-2010-024, ISSN 0012-7094, S2CID 9262012

semanticscholar.org

api.semanticscholar.org

Harvey, David (2010), "A multimodular algorithm for computing Bernoulli numbers", Math. Comput., 79 (272): 2361–2370, arXiv:0807.1347, doi:10.1090/S0025-5718-2010-02367-1, S2CID 11329343, Zbl 1215.11016
Guo, Victor J. W.; Zeng, Jiang (30 August 2005), "A q-Analogue of Faulhaber's Formula for Sums of Powers", The Electronic Journal of Combinatorics, 11 (2), arXiv:math/0501441, Bibcode:2005math......1441G, doi:10.37236/1876, S2CID 10467873
Stanley, Richard P. (2010), "A survey of alternating permutations", Combinatorics and graphs, Contemporary Mathematics, vol. 531, Providence, RI: American Mathematical Society, pp. 165–196, arXiv:0912.4240, doi:10.1090/conm/531/10466, ISBN 978-0-8218-4865-4, MR 2757798, S2CID 14619581
Charollois, Pierre; Sczech, Robert (2016), "Elliptic Functions According to Eisenstein and Kronecker: An Update", EMS Newsletter, 2016–9 (101): 8–14, doi:10.4171/NEWS/101/4, ISSN 1027-488X, S2CID 54504376
Bannai, Kenichi; Kobayashi, Shinichi (2010), "Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers", Duke Mathematical Journal, 153 (2), arXiv:math/0610163, doi:10.1215/00127094-2010-024, ISSN 0012-7094, S2CID 9262012
Agoh, Takashi; Dilcher, Karl (2008), "Reciprocity Relations for Bernoulli Numbers", American Mathematical Monthly, 115 (3): 237–244, doi:10.1080/00029890.2008.11920520, JSTOR 27642447, S2CID 43614118

stanford.edu

www-cs-faculty.stanford.edu

Donald Knuth (2022), Recent News (2022): Concrete Mathematics and Bernoulli.
But last year I took a close look at Peter Luschny's Bernoulli manifesto, where he gives more than a dozen good reasons why the value of $B_1$ should really be plus one-half. He explains that some mathematicians of the early 20th century had unilaterally changed the conventions, because some of their formulas came out a bit nicer when the negative value was used. It was their well-intentioned but ultimately poor choice that had led to what I'd been taught in the 1950s. […] By now, hundreds of books that use the “minus-one-half” convention have unfortunately been written. Even worse, all the major software systems for symbolic mathematics have that 20th-century aberration deeply embedded. Yet Luschny convinced me that we have all been wrong, and that it's high time to change back to the correct definition before the situation gets even worse.

tripod.com

jeff560.tripod.com

The Mathematics Genealogy Project (n.d.) shows Leibniz as the academic advisor of Jakob Bernoulli. See also Miller (2017). The Mathematics Genealogy Project, Fargo: Department of Mathematics, North Dakota State University, n.d., archived from the original on 10 May 2019, retrieved 11 May 2019. Miller, Jeff (23 June 2017), "Earliest Uses of Symbols of Calculus", Earliest Uses of Various Mathematical Symbols, retrieved 11 May 2019.

univie.ac.at

mat.univie.ac.at

Dumont, D. (1981), "Matrices d'Euler-Seidel", Séminaire Lotharingien de Combinatoire, B05c

web.archive.org

The Mathematics Genealogy Project (n.d.) shows Leibniz as the academic advisor of Jakob Bernoulli. See also Miller (2017). The Mathematics Genealogy Project, Fargo: Department of Mathematics, North Dakota State University, n.d., archived from the original on 10 May 2019, retrieved 11 May 2019. Miller, Jeff (23 June 2017), "Earliest Uses of Symbols of Calculus", Earliest Uses of Various Mathematical Symbols, retrieved 11 May 2019.

wolfram.com

mathworld.wolfram.com

Weisstein, Eric W., "Bernoulli Number", MathWorld

blog.wolfram.com

Pavlyk, Oleksandr (29 April 2008), "Today We Broke the Bernoulli Record: From the Analytical Engine to Mathematica", Wolfram News.

worldcat.org

search.worldcat.org

Kitagawa, Tomoko L. (2021-07-23), "The Origin of the Bernoulli Numbers: Mathematics in Basel and Edo in the Early Eighteenth Century", The Mathematical Intelligencer, 44: 46–56, doi:10.1007/s00283-021-10072-y, ISSN 0343-6993
Charollois, Pierre; Sczech, Robert (2016), "Elliptic Functions According to Eisenstein and Kronecker: An Update", EMS Newsletter, 2016–9 (101): 8–14, doi:10.4171/NEWS/101/4, ISSN 1027-488X, S2CID 54504376
Bannai, Kenichi; Kobayashi, Shinichi (2010), "Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers", Duke Mathematical Journal, 153 (2), arXiv:math/0610163, doi:10.1215/00127094-2010-024, ISSN 0012-7094, S2CID 9262012

zbmath.org

Harvey, David (2010), "A multimodular algorithm for computing Bernoulli numbers", Math. Comput., 79 (272): 2361–2370, arXiv:0807.1347, doi:10.1090/S0025-5718-2010-02367-1, S2CID 11329343, Zbl 1215.11016
Neukirch, Jürgen (1999), Algebraische Zahlentheorie, Grundlehren der mathematischen Wissenschaften, vol. 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021 §VII.2.

zenodo.org

Jacobi, C.G.J. (1834), "De usu legitimo formulae summatoriae Maclaurinianae", Journal für die reine und angewandte Mathematik, 12: 263–272