Bernoulli polynomials explained

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.

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

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

Generating functions

The generating function for the Bernoulli polynomials is$$\backslash frac=\; \backslash sum\_^\backslash infty\; B\_n(x)\; \backslash frac.$$The generating function for the Euler polynomials is$$\backslash frac=\; \backslash sum\_^\backslash infty\; E\_n(x)\; \backslash frac.$$

Explicit formula

$$B\_n(x)\; =\; \backslash sum\_^n\; B\_\; x^k,$$$$E\_m(x)=\backslash sum\_^m\; \backslash frac\backslash left(x-\backslash tfrac12\backslash right)^\; .$$for n ≥ 0, where B_k are the Bernoulli numbers, and E_k are the Euler numbers.

Representation by a differential operator

The Bernoulli polynomials are also given by$$B\_n(x)\; =\; \backslash frac\; x^n$$where D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series. It follows that $$\backslash int\_a^x\; B\_n\; (u)\backslash ,du\; =\; \backslash frac.$$cf. below. By the same token, the Euler polynomials are given by$$E\_n(x)\; =\; \backslash frac\; x^n.$$

Representation by an integral operator

The Bernoulli polynomials are also the unique polynomials determined by$$\backslash int\_x^\; B\_n(u)\backslash ,du\; =\; x^n.$$

The integral transform$$(Tf)(x)\; =\; \backslash int\_x^\; f(u)\backslash ,du$$on polynomials f, simply amounts to This can be used to produce the inversion formulae below.

Integral Recurrence

In,^{[1] [2]} it is deduced and proved that the Bernoulli polynomials can be obtained by the following integral recurrence$$B\_(x)=m\; \backslash int\_^\; B\_(t)\backslash ,dt-m\backslash int\_^\; \backslash int\_0^t\; B\_(s)\backslash ,ds\; dt.$$

Another explicit formula

An explicit formula for the Bernoulli polynomials is given by$$B\_n(x)\; =\; \backslash sum\_^n\; \backslash biggl[\backslash frac\{1\}\{k\; +\; 1\}\; \backslash sum\_\{\backslash ell=0\}^k\; (-1)^\backslash ell\; \{\; k\; \backslash choose\; \backslash ell\; \}\; (x\; +\; \backslash ell)^n\; \backslash biggr].$$

That is similar to the series expression for the Hurwitz zeta function in the complex plane. Indeed, there is the relationship$$B\_n(x)\; =\; -n\; \backslash zeta(1\; -\; n,\backslash ,x)$$where

\zeta(s,q) is the Hurwitz zeta function. The latter generalizes the Bernoulli polynomials, allowing for non-integer values

The inner sum may be understood to be the th forward difference of

x ^{m,} that is,$$\backslash Delta^n\; x^m\; =\; \backslash sum\_^n\; (-1)^(x\; +\; k)^m$$where \Delta is the forward difference operator. Thus, one may write$$B\_n(x)\; =\; \backslash sum\_^n\; \backslash frac\backslash Delta^k\; x^n.$$

This formula may be derived from an identity appearing above as follows. Since the forward difference operator equals$$\backslash Delta\; =\; e^D\; -\; 1$$where is differentiation with respect to, we have, from the Mercator series,$$\backslash frac\; =\; \backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash frac.$$

As long as this operates on an th-degree polynomial such as

x ^{m,} one may let go from only up

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$$E\_n(x)\; =\; \backslash sum\_^n\; \backslash left[\backslash frac\{1\}\{2^k\}\backslash sum\_\{\backslash ell=0\}^n\; (-1)^\backslash ell\; \{k\; \backslash choose\; \backslash ell\}(x\; +\; \backslash ell)^n\; \backslash right]\; .$$

The above follows analogously, using the fact that$$\backslash frac\; =\; \backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash bigl(\backslash Delta\; \backslash bigr)^n\; .$$

Sums of pth powers

See main article: Faulhaber's formula.

Using either the above integral representation of

x ^{n} or the identity B _{n(x} + 1 ) - B _{n(x)} = nx ^{n-1} , we have$$\backslash sum\_^x\; k^p\; =\; \backslash int\_0^\; B\_p(t)\; \backslash ,\; dt\; =\; \backslash frac$$(assuming 0⁰ = 1).

Explicit expressions for low degrees

The first few Bernoulli polynomials are:

The first few Euler polynomials are:

Maximum and minimum

At higher the amount of variation in

B _{n(x)} between x = 0 and x = 1 gets large. For instance, B _{16} ( 0 ) = B _{16} ( 1 ) = {} - \tfrac{ 3617 }{ 510 } ≈ -7 . 09 , but B _{16} l (\tfrac 12 r ) = {} \tfrac{ 118518239 }{ 3342336 } ≈ 7 . 09 . showed that the maximum value of B _{n(x)} between and obeysunless is in which case$$M\_n\; =\; \backslash frac$$(where \zeta(x) is the Riemann zeta function), while the minimum obeys$$m\_n\; >\; \backslash frac$$unless in which case$$m\_n\; =\; \backslash frac.$$

These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well.

Differences and derivatives

The Bernoulli and Euler polynomials obey many relations from umbral calculus:(is the forward difference operator). Also,$$E\_n(x+1)\; +\; E\_n(x)\; =\; 2x^n.$$These polynomial sequences are Appell sequences:

Translations

These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials are another example.)

Symmetries

Zhi-Wei Sun and Hao Pan ^[3] established the following surprising symmetry relation: If and, then$$r[s,t;x,y]\_n+s[t,r;y,z]\_n+t[r,s;z,x]\_n=0,$$where$$[s,t;x,y]\_n=\backslash sum\_^n(-1)^k\; B\_(x)B\_k(y).$$

Fourier series

The Fourier series of the Bernoulli polynomials is also a Dirichlet series, given by the expansion$$B\_n(x)\; =\; -\backslash frac\backslash sum\_\backslash frac=\; -2\; n!\; \backslash sum\_^\; \backslash frac.$$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$$B\_n(x)\; =\; -\backslash Gamma(n+1)\; \backslash sum\_^\backslash infty\; \backslash frac\; .$$

This expansion is valid only for when and is valid for when .

The Fourier series of the Euler polynomials may also be calculated. Defining the functionsfor

\nu > 1 , the Euler polynomial has the Fourier seriesNote that the C _{\nu} and S _{\nu} are odd and even, respectively:\chi _{\nu} as

Inversion

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 $$x^n\; =\; \backslash frac\; \backslash sum\_^n\; B\_k\; (x)$$and$$x^n\; =\; E\_n\; (x)\; +\; \backslash frac\; \backslash sum\_^\; E\_k\; (x).$$

Relation to falling factorial

(x) _{k} as$$B\_(x)\; =\; B\_\; +\; \backslash sum\_^n\backslash frac\backslash left\backslash (x)\_$$where B _{n} = B _{n(0)} and$$\backslash left\backslash \; =\; S(n,k)$$denotes the Stirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:$$(x)\_\; =\; \backslash sum\_^n\backslash frac\backslash left[\backslash begin\{matrix\}\; n\; \backslash \backslash \; k\; \backslash end\{matrix\}\; \backslash right]\backslash left(B\_(x)\; -\; B\_\; \backslash right)$$where$$\backslash left[\backslash begin\{matrix\}\; n\; \backslash \backslash \; k\; \backslash end\{matrix\}\; \backslash right]\; =\; s(n,k)$$denotes the Stirling number of the first kind.

Multiplication theorems

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

For a natural number,$$B\_n(mx)=\; m^\; \backslash sum\_^\; B\_n$$

Integrals

Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:^[4]

\int ^{1} _{0} B _{n(t)} B _{m(t)dt} = ( -1 ) ^{n-1} (m! n!)/((m + n)!) B _{n+m} for m,n \geq 1 \int ^{1} _{0} E _{n(t)} E _{m(t)dt} = ( -1 ) ^{n} 4 ( 2 ^{m+n+2} -1 ) (m!n!)/((m + n +2 )!) B _{n+m+2}

Another integral formula states^[5]

\int ^{1} _{0} E _{n} \left( x + y\right) log (\tan (\pi)/(2) x)dx = n! \sum ^{\left\lfloor (n +1)/(2 \right\rfloor)} _{k =1} (( -1 ) ^{k-1})/(\pi ^{2k}) \left( 2-2 ^{-2k} \right)\zeta( 2 k +1 ) (y ^{n+1-2k})/((n +1-2 k)!) with the special case for y =0 \int ^{1} _{0} E _{2n-1} \left( x \right) log (\tan (\pi)/(x)dx = 2) (( -1 ) ^{n-1} ( 2 n -1 )!)/(\pi ^{2n}) \left( 2-2 ^{-2n} \right)\zeta( 2 n +1 ) \int ^{1} _{0} B _{2n-1} \left( x \right) log (\tan (\pi)/(x)dx = 2) (( -1 ) ^{n-1})/(\pi ^{2n}) (2 ^{2n-2})/(( 2 n -1 )!) \sum ^{n} _{k =1} ( 2 ^{2k+1} -1 )\zeta( 2 k +1 )\zeta( 2 n -2 k) \int ^{1} _{0} E _{2n} \left( x \right) log (\tan (\pi)/(2) x)dx = \int ^{1} _{0} B _{2n} \left( x \right) log (\tan (\pi)/(2) x)dx =0 \int ^{1} _{0} {{{B} _{2n-1} }\left(x \right)\cot \left(\pi x \right)dx}=\frac\zeta \left(2n-1 \right)

Periodic Bernoulli polynomials

A periodic Bernoulli polynomial is a Bernoulli polynomial evaluated at the fractional part of the argument . 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 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

x :P _{k(x)} is continuous for all k > 1 P _{k'(x)} exists and is continuous for k > 2 P' _{k(x)} = k P _{k-1} (x) for k > 2

See also

• Bernoulli numbers
• Bernoulli polynomials of the second kind
• Stirling polynomial
• Polynomials calculating sums of powers of arithmetic progressions

References

• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (1972) Dover, New York. (See Chapter 23)
• (See chapter 12.11)
  • Cvijović . Djurdje . Klinowski . Jacek . 1995 . New formulae for the Bernoulli and Euler polynomials at rational arguments . . 123 . 5 . 1527–1535 . 10.1090/S0002-9939-1995-1283544-0 . free . 2161144 .
• 10.1007/s11139-007-9102-0 . Guillera . Jesus . Sondow . Jonathan . 2008 . Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent . math.NT/0506319 . The Ramanujan Journal . 16 . 3. 247–270 . 14910435 . (Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)
• Book: Hugh L. Montgomery . Hugh Montgomery (mathematician) . Robert C. Vaughan . Robert Charles Vaughan (mathematician) . Multiplicative number theory I. Classical theory . Cambridge tracts in advanced mathematics . 97 . 2007 . 978-0-521-84903-6 . 495–519 . Cambridge Univ. Press . Cambridge .

External links

• A list of integral identities involving Bernoulli polynomials from NIST

Notes and References

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. Zhi-Wei Sun . Hao Pan . Acta Arithmetica . 125 . 2006 . 21–39 . Identities concerning Bernoulli and Euler polynomials . 1 . math/0409035 . 10.4064/aa125-1-3. 2006AcAri.125...21S . 10841415 .
4. amp . Takashi Agoh . Karl Dilcher . Journal of Mathematical Analysis and Applications . 381 . 2011 . 10–16 . Integrals of products of Bernoulli polynomials . 10.1016/j.jmaa.2011.03.061 . free .
5. Elaissaoui, Lahoucine . Guennoun, Zine El Abidine . amp . Evaluation of log-tangent integrals by series involving ζ(2n+1). Integral Transforms and Special Functions . English . 2017. 28 . 6 . 460–475 . 10.1080/10652469.2017.1312366 . 1611.01274 . 119132354 .