Bernoulli%20and%20Euler%20polynomials
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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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The real zeros of the Bernoulli polynomials.
J. Approx. Theory 58 (2), pp. 124–150.
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Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials.
J. Math. Anal. Appl. 239 (2), pp. 457–477.
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Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions.
Stud. Appl. Math. 103 (3), pp. 241–258.
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Large degree asymptotics of generalized Bernoulli and Euler polynomials.
J. Math. Anal. Appl. 363 (1), pp. 197–208.
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2: 25.6 Integer Arguments
3: 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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Integrals of products of Bernoulli polynomials.
J. Math. Anal. Appl. 381 (1), pp. 10–16.
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Some determinants of Bernoulli, Euler and related numbers.
Portugal. Math. 18, pp. 91–99.
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Bernoulli’s power-sum formulas revisited.
Math. Gaz. 90 (518), pp. 276–279.
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A primer on Bernoulli numbers and polynomials.
Math. Mag. 81 (3), pp. 178–190.
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4: 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
… ► …5: Software Index
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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5.24(ii) , | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | FDLIBM |
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18 Orthogonal Polynomials | |||||||||||||||||||||||||
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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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6: 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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A note on Euler numbers and polynomials.
Nagoya Math. J. 7, pp. 35–43.
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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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7: 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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On the degree of an irreducible factor of the Bernoulli polynomials.
Acta Arith. 50 (3), pp. 243–249.
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Computation of tangent, Euler, and Bernoulli numbers.
Math. Comp. 21 (100), pp. 663–688.
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8: 5.11 Asymptotic Expansions
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►For the Bernoulli numbers , see §24.2(i).
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►The scaled gamma function is defined in (5.11.3) and its main property is as in the sector .
Wrench (1968) gives exact values of up to .
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►where is fixed, and is the Bernoulli polynomial defined in §24.2(i).
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►In terms of generalized Bernoulli polynomials
(§24.16(i)), we have for ,
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9: 25.11 Hurwitz Zeta Function
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§25.11(iii) Representations by the Euler–Maclaurin Formula
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25.11.6
, , .
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25.11.7
, , , .
►For see §24.2(iii).
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25.11.14
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10: Bibliography W
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Prime Divisors of the Bernoulli and Euler Numbers.
In Number Theory for the Millennium, III (Urbana, IL, 2000),
pp. 357–374.
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The zeros of Euler’s psi function and its derivatives.
J. Math. Anal. Appl. 332 (1), pp. 607–616.
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Global asymptotics of the Meixner polynomials.
Asymptotic Analysis 75 (3-4), pp. 211–231.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials.
Ph.D. Thesis, University of Wisconsin, Madison, WI.
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