The Bernoulli polynomials are also given by
where is differentiation with respect to x and the fraction is expanded as a formal power series. It follows that
cf. § Integrals below. By the same token, the Euler polynomials are given by
Representation by an integral operator
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The Bernoulli polynomials are also the unique polynomials determined by
In,[1][2] it is deduced and proved that the Bernoulli polynomials can be obtained by the following integral recurrence
Another explicit formula
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An explicit formula for the Bernoulli polynomials is given by
That is similar to the series expression for the Hurwitz zeta function in the complex plane. Indeed, there is the relationship
where is the Hurwitz zeta function. The latter generalizes the Bernoulli polynomials, allowing for non-integer values of n.
This formula may be derived from an identity appearing above as follows. Since the forward difference operator Δ equals
where D is differentiation with respect to x, we have, from the Mercator series,
As long as this operates on an mth-degree polynomial such as one may let n go from 0 only up to m.
An integral representation for the Bernoulli polynomials is given by the Nörlund–Rice integral, which follows from the expression as a finite difference.
An explicit formula for the Euler polynomials is given by
The above follows analogously, using the fact that
At higher n the amount of variation in between and gets large. For instance, but Lehmer (1940)[3] showed that the maximum value (Mn) of between 0 and 1 obeys
unless n is 2 modulo 4, in which case
(where is the Riemann zeta function), while the minimum (mn) obeys
unless n = 0 modulo 4 , in which case
These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well.
These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials are another example.)
Symmetries
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Zhi-Wei Sun and Hao Pan [4] established the following surprising symmetry relation: If r + s + t = n and x + y + z = 1, then
where
Fourier series
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The Fourier series of the Bernoulli polynomials is also a Dirichlet series, given by the expansion
Note the simple large n limit to suitably scaled trigonometric functions.
This expansion is valid only for 0 ≤ x ≤ 1 when n ≥ 2 and is valid for 0 < x < 1 when n = 1.
The Fourier series of the Euler polynomials may also be calculated. Defining the functions
for , the Euler polynomial has the Fourier series
Note that the and are odd and even, respectively:
Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions, and P0(x) is not even a function, being the derivative of a sawtooth and so a Dirac comb.
The following properties are of interest, valid for all :
^Hurtado Benavides, Miguel Ángel. (2020). De las sumas de potencias a las sucesiones de Appell y su caracterización a través de funcionales. [Tesis de maestría]. Universidad Sergio Arboleda. https://repository.usergioarboleda.edu.co/handle/11232/174
^Sergio A. Carrillo; Miguel Hurtado. Appell and Sheffer sequences: on their characterizations through functionals and examples. Comptes Rendus. Mathématique, Tome 359 (2021) no. 2, pp. 205-217. doi : 10.5802/crmath.172. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.172/
^Takashi Agoh & Karl Dilcher (2011). "Integrals of products of Bernoulli polynomials". Journal of Mathematical Analysis and Applications. 381: 10–16. doi:10.1016/j.jmaa.2011.03.061.
^Elaissaoui, Lahoucine & Guennoun, Zine El Abidine (2017). "Evaluation of log-tangent integrals by series involving ζ(2n+1)". Integral Transforms and Special Functions. 28 (6): 460–475. arXiv:1611.01274. doi:10.1080/10652469.2017.1312366. S2CID 119132354.
Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (1972) Dover, New York. (See Chapter 23)
Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 (See chapter 12.11)
Cvijović, Djurdje; Klinowski, Jacek (1995). "New formulae for the Bernoulli and Euler polynomials at rational arguments". Proceedings of the American Mathematical Society. 123 (5): 1527–1535. doi:10.1090/S0002-9939-1995-1283544-0. JSTOR 2161144.
Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". The Ramanujan Journal. 16 (3): 247–270. arXiv:math.NT/0506319. doi:10.1007/s11139-007-9102-0. S2CID 14910435. (Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)
Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge: Cambridge Univ. Press. pp. 495–519. ISBN 978-0-521-84903-6.
External links
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A list of integral identities involving Bernoulli polynomials from NIST