Bernoulli%20numbers
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1: Bibliography L
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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2: 25.6 Integer Arguments
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25.6.3
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3: 24.2 Definitions and Generating Functions
§24.2 Definitions and Generating Functions
►§24.2(i) Bernoulli Numbers and Polynomials
… ►§24.2(ii) Euler Numbers and Polynomials
… ►§24.2(iii) Periodic Bernoulli and Euler Functions
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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A Classical Introduction to Modern Number Theory.
2nd edition, Springer-Verlag, New York.
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5: Software Index
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Software Associated with Books.
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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20 Theta Functions | |||||||||||||||||||||||||
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24 Bernoulli and Euler Polynomials | |||||||||||||||||||||||||
24.21(ii) , , , | ✓ | ✓ | ✓ | ✓ | a | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | Derive, MuPAD | ||||||||||||
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27 Functions of Number Theory | |||||||||||||||||||||||||
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An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.
6: 24.20 Tables
§24.20 Tables
… ►Wagstaff (1978) gives complete prime factorizations of and for and , respectively. …7: Bibliography
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Congruences of -adic integer order Bernoulli numbers.
J. Number Theory 59 (2), pp. 374–388.
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Some determinants of Bernoulli, Euler and related numbers.
Portugal. Math. 18, pp. 91–99.
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Sharp bounds for the Bernoulli numbers.
Arch. Math. (Basel) 74 (3), pp. 207–211.
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A primer on Bernoulli numbers and polynomials.
Math. Mag. 81 (3), pp. 178–190.
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8: Bibliography C
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Some congruences for the Bernoulli numbers.
Amer. J. Math. 75 (1), pp. 163–172.
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-Bernoulli and Eulerian numbers.
Trans. Amer. Math. Soc. 76 (2), pp. 332–350.
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Expansions of -Bernoulli numbers.
Duke Math. J. 25 (2), pp. 355–364.
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Accélération de calcul de nombres de Bernoulli.
J. Number Theory 28 (3), pp. 347–362 (French).
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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9: Bibliography K
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Poly-Bernoulli numbers.
J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.
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The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers.
Math. Ann. 216 (1), pp. 1–4.
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On formulas involving both the Bernoulli and Fibonacci numbers.
Scripta Math. 23, pp. 27–35.
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Remark on -adic -Bernoulli numbers.
Adv. Stud. Contemp. Math. (Pusan) 1, pp. 127–136.
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Computation of tangent, Euler, and Bernoulli numbers.
Math. Comp. 21 (100), pp. 663–688.
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10: 26.14 Permutations: Order Notation
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►As an example, is an element of The inversion number is the number of pairs of elements for which the larger element precedes the smaller:
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►The Eulerian number, denoted , is the number of permutations in with exactly descents.
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