Bernoulli polynomials

Summary

In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula.

Bernoulli polynomials

These polynomials occur in the study of many special functions and, in particular, the Riemann zeta function and the Hurwitz zeta function. They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). For the Bernoulli polynomials, the number of crossings of the x-axis in the unit interval does not go up with the degree. In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions.

A similar set of polynomials, based on a generating function, is the family of Euler polynomials.

Representations

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The Bernoulli polynomials Bn can be defined by a generating function. They also admit a variety of derived representations.

Generating functions

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The generating function for the Bernoulli polynomials is   The generating function for the Euler polynomials is  

Explicit formula

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    for n ≥ 0, where Bk are the Bernoulli numbers, and Ek are the Euler numbers.

Representation by a differential operator

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The Bernoulli polynomials are also given by   where D = d/dx 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  

The integral transform   on polynomials f, simply amounts to   This can be used to produce the inversion formulae below.

Integral Recurrence

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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.

The inner sum may be understood to be the nth forward difference of   that is,   where   is the forward difference operator. Thus, one may write  

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  

Sums of pth powers

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Using either the above integral representation of   or the identity  , we have   (assuming 00 = 1).

The Bernoulli and Euler numbers

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The Bernoulli numbers are given by  

This definition gives   for  

An alternate convention defines the Bernoulli numbers as  

The two conventions differ only when   since  

The Euler numbers are given by  

Explicit expressions for low degrees

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The first few Bernoulli polynomials are:  

The first few Euler polynomials are:  

Maximum and minimum

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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.

Differences and derivatives

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The Bernoulli and Euler polynomials obey many relations from umbral calculus:   (Δ is the forward difference operator). Also,   These polynomial sequences are Appell sequences:  

Translations

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  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 is a special case of the analogous form for the Hurwitz zeta function  

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: 

They are related to the Legendre chi function   as  

Inversion

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The Bernoulli and Euler polynomials may be inverted to express the monomial in terms of the polynomials.

Specifically, evidently from the above section on integral operators, it follows that   and  

Relation to falling factorial

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The Bernoulli polynomials may be expanded in terms of the falling factorial   as   where   and   denotes the Stirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:   where   denotes the Stirling number of the first kind.

Multiplication theorems

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The multiplication theorems were given by Joseph Ludwig Raabe in 1851:

For a natural number m≥1,    

Integrals

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Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:[5]

  •  
  •  

Another integral formula states[6]

  •  

with the special case for  

  •  
  •  
  •  
  •  

Periodic Bernoulli polynomials

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A periodic Bernoulli polynomial Pn(x) is a Bernoulli polynomial evaluated at the fractional part of the argument x. These functions are used to provide the remainder term in the Euler–Maclaurin formula relating sums to integrals. The first polynomial is a sawtooth function.

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  :

  •   is continuous for all  
  •   exists and is continuous for  
  •   for  

See also

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References

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  1. ^ 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
  2. ^ 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/
  3. ^ Lehmer, D.H. (1940). "On the maxima and minima of Bernoulli polynomials". American Mathematical Monthly. 47: 533–538.
  4. ^ Zhi-Wei Sun; Hao Pan (2006). "Identities concerning Bernoulli and Euler polynomials". Acta Arithmetica. 125 (1): 21–39. arXiv:math/0409035. Bibcode:2006AcAri.125...21S. doi:10.4064/aa125-1-3. S2CID 10841415.
  5. ^ 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.
  6. ^ 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)
  • Dilcher, K. (2010), "Bernoulli and Euler Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
  • 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.
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  • A list of integral identities involving Bernoulli polynomials from NIST