Bernoulli polynomials

[[File:Bernoulli polynomials.svg|thumb|right|Bernoulli polynomials]]

In [[mathematics]], the '''Bernoulli polynomials''', named after [[Jacob Bernoulli]], combine the [[Bernoulli number]]s and [[binomial coefficient]]s. They are used for [[series expansion]] of [[function (mathematics)|functions]], and with the [[Euler–MacLaurin formula]].

These [[polynomial]]s 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 of a polynomial|degree]]. In the limit of large degree, they approach, when appropriately scaled, the [[trigonometric function|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 [[exponential generating function]] for the Bernoulli polynomials is <math display="block">\frac{t e^{xt}}{e^t-1}= \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}.</math> The exponential generating function for the Euler polynomials is <math display="block">\frac{2 e^{xt}}{e^t+1}= \sum_{n=0}^\infty E_n(x) \frac{t^n}{n!}.</math>

===Explicit formula===

<math display="block">B_n(x) = \sum_{k=0}^n {n \choose k} B_{n-k} x^k,</math> <math display="block">E_m(x)= \sum_{k=0}^m {m \choose k} \frac{E_k}{2^k} \left(x-\tfrac12\right)^{m-k} .</math> for <math>n \geq 0</math>, where <math>B_k</math> are the [[Bernoulli number]]s, and <math>E_k</math> are the [[Euler numbers]]. It follows that <math>B_n(0) = B_n</math> and <math>E_m\big(\tfrac{1}{2}\big) = \tfrac{1}{2^m} E_m</math>.

===Representation by a differential operator===

The Bernoulli polynomials are also given by <math display="block">\ B_n(x) = \frac{ D }{\ e^D -1\ }\ x^n\ </math> where <math>\ D \equiv \frac{ \mathrm{d} }{\ \mathrm{d} x\ }\ </math> is differentiation with respect to {{mvar|x}} and the fraction is expanded as a [[formal power series]]. It follows that <math display="block">\ \int_a^x\ B_n(u)\ \mathrm{d}\ u = \frac{\ B_{n+1}(x) - B_{n+1}(a)\ }{ n + 1 } ~.</math> cf. {{slink||Integrals}} below. By the same token, the Euler polynomials are given by <math display="block">\ E_n(x) = \frac{ 2 }{\ e^D + 1\ }\ x^n ~.</math>

===Representation by an integral operator===

The Bernoulli polynomials are also the unique polynomials determined by <math display="block">\int_x^{x+1} B_n(u)\,du = x^n.</math>

The [[integral transform]] <math display="block">(Tf)(x) = \int_x^{x+1} f(u)\,du</math> on polynomials ''f'', simply amounts to <math display="block">\begin{align} (Tf)(x) = {e^D - 1 \over D}f(x) & {} = \sum_{n=0}^\infty {D^n \over (n+1)!}f(x) \\ & {} = f(x) + {f'(x) \over 2} + {f''(x) \over 6} + {f'''(x) \over 24} + \cdots . \end{align}</math> This can be used to produce the [[#Inversion|inversion formulae below]].

=== Integral recurrence ===

In,<ref>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</ref><ref>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/</ref> it is deduced and proved that the Bernoulli polynomials can be obtained by the following integral recurrence <math display="block">B_{m}(x)=m \int_{0}^{x} B_{m-1}(t)\,dt-m\int_{0}^{1} \int_0^t B_{m-1}(s)\,ds dt.</math>

===Another explicit formula===

An explicit formula for the Bernoulli polynomials is given by <math display="block"> B_n(x) = \sum_{k=0}^n \biggl[ \frac{1}{k + 1} \sum_{\ell=0}^k (-1)^\ell { k \choose \ell } (x + \ell)^n \biggr].</math>

That is similar to the series expression for the [[Hurwitz zeta function]] in the [[complex plane]]. Indeed, there is the relationship <math display="block">B_n(x) = -n \zeta(1 - n,\,x)</math> where <math>\zeta(s,\,q)</math> is the [[Hurwitz zeta function]]. The latter generalizes the Bernoulli polynomials, allowing for non-integer values {{nobr|of {{mvar|n}}.}}

The inner sum may be understood to be the {{mvar|n}}th [[forward difference]] of <math>x^m,</math> that is, <math display="block">\Delta^n x^m = \sum_{k=0}^n (-1)^{n - k}{n \choose k}(x + k)^m</math> where <math>\Delta</math> is the [[forward difference operator]]. Thus, one may write <math display="block">B_n(x) = \sum_{k=0}^n \frac{(-1)^k}{k + 1}\Delta^k x^n.</math>

This formula may be derived from an identity appearing above as follows. Since the forward difference operator {{math|Δ}} equals <math display="block">\Delta = e^D - 1</math> where {{mvar|D}} is differentiation with respect to {{mvar|x}}, we have, from the [[Mercator series]], <math display="block">\frac{ D }{e^D - 1} = \frac{\log(\Delta + 1)}{\Delta} = \sum_{n=0}^\infty \frac{(-\Delta)^n }{n + 1}.</math>

As long as this operates on an {{mvar|m}}th-degree polynomial such as <math>x^m,</math> one may let {{mvar|n}} go from {{math|0}} only up {{nobr|to {{mvar|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 <math display="block">E_n(x) = \sum_{k=0}^n \left[ \frac{1}{2^k}\sum_{\ell=0}^n (-1)^\ell {k \choose \ell}(x + \ell)^n \right] .</math>

The above follows analogously, using the fact that <math display="block">\frac{2}{e^D + 1} = \frac{1}{1 + \tfrac12 \Delta} = \sum_{n = 0}^\infty \bigl( {-\tfrac{1}{2}} \Delta \bigr)^n .</math>

==Sums of ''p''th powers== {{main|Faulhaber's formula}}

Using either the above [[#Representation by an integral operator|integral representation]] of <math>x^n</math> or the [[#Differences and derivatives|identity]] <math> B_n(x + 1) - B_n(x) = nx^{n-1}</math>, we have <math display="block">\sum_{k=0}^x k^p = \int_0^{x+1} B_p(t) \, dt = \frac{B_{p+1}(x+1)-B_{p+1}}{p+1} </math> (assuming 00 = 1).

==Explicit expressions for low degrees== The first few Bernoulli polynomials are: <math display="block"> \begin{align} B_0(x) & = 1, & B_4(x) & = x^4 - 2x^3 + x^2 - \tfrac{1}{30}, \\[4mu] B_1(x) & = x - \tfrac{1}{2}, & B_5(x) & = x^5 - \tfrac{5}{2}x^4 + \tfrac{5}{3}x^3 - \tfrac{1}{6}x, \\[4mu] B_2(x) & = x^2 - x + \tfrac{1}{6}, & B_6(x) & = x^6 - 3x^5 + \tfrac{5}{2}x^4 - \tfrac{1}{2}x^2 + \tfrac{1}{42}, \\[-2mu] B_3(x) & = x^3 - \tfrac{3}{2}x^2 + \tfrac{1}{2}x \vphantom\Big|, \qquad & &\ \,\, \vdots \end{align} </math>

The first few Euler polynomials are: <math display="block"> \begin{align} E_0(x) & = 1, & E_4(x) & = x^4 - 2x^3 + x, \\[4mu] E_1(x) & = x - \tfrac{1}{2}, & E_5(x) & = x^5 - \tfrac{5}{2}x^4 + \tfrac{5}{2}x^2 - \tfrac{1}{2}, \\[4mu] E_2(x) & = x^2 - x, & E_6(x) & = x^6 - 3x^5 + 5x^3 - 3x, \\[-1mu] E_3(x) & = x^3 - \tfrac{3}{2}x^2 + \tfrac{1}{4}, \qquad \ \ & &\ \,\, \vdots \end{align} </math>

==Maximum and minimum==

At higher {{mvar|n}} the amount of variation in <math>B_n(x)</math> between <math>x = 0</math> and <math>x = 1</math> gets large. For instance, <math>B_{16}(0) = B_{16}(1) = {}</math><math> -\tfrac{3617}{510} \approx -7.09,</math> but <math>B_{16}\bigl(\tfrac12\bigr) = {}</math><math>\tfrac{118518239}{3342336} \approx 7.09.</math> {{nobr|[[D.H. Lehmer|Lehmer]] (1940)<ref>{{cite journal |first=D.H. |last=Lehmer |author-link=D.H. Lehmer |year=1940 |title=On the maxima and minima of Bernoulli polynomials |journal=[[American Mathematical Monthly]] |volume=47 |issue=8 |pages=533–538 |doi=10.1080/00029890.1940.11991015 }}</ref>}} showed that the maximum value ({{mvar|M{{sub|n}}}}) of <math>B_n(x)</math> between {{math|0}} and {{math|1}} obeys <math display="block">M_n < \frac{2n!}{(2\pi)^n}</math> unless {{mvar|n}} is {{nobr|{{math|2 modulo 4}},}} in which case <math display="block">M_n = \frac{2\zeta (n)\,n!}{(2\pi)^n}</math> (where <math>\zeta(x)</math> is the [[Riemann zeta function]]), while the minimum ({{mvar|m{{sub|n}}}}) obeys <math display="block">m_n > \frac{ -2 n!}{(2\pi)^n}</math> unless {{nobr| {{math|1=''n'' = 0 modulo 4 }} ,}} in which case <math display="block">m_n = \frac{-2 \zeta(n)\,n! }{(2\pi)^n}.</math>

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]]: <math display="block">\begin{align} \Delta B_n(x) &= B_n(x+1)-B_n(x)=nx^{n-1}, \\[3mu] \Delta E_n(x) &= E_n(x+1)-E_n(x)=2(x^n-E_n(x)). \end{align}</math> ({{math|Δ}} is the [[forward difference operator]]). Also, <math display="block"> E_n(x+1) + E_n(x) = 2x^n.</math> These [[polynomial sequence]]s are [[Appell sequence]]s: <math display="block">\begin{align} B_n'(x) &= n B_{n-1}(x), \\[3mu] E_n'(x) &= n E_{n-1}(x). \end{align}</math>

===Translations===

<math display="block">\begin{align} B_n(x+y) &= \sum_{k=0}^n {n \choose k} B_k(x) y^{n-k} \\[3mu] E_n(x+y) &= \sum_{k=0}^n {n \choose k} E_k(x) y^{n-k} \end{align}</math> These identities are also equivalent to saying that these polynomial sequences are [[Appell sequence]]s. ([[Hermite polynomials]] are another example.)

===Symmetries===

<math display="block">\begin{align} B_n(1-x) &= \left(-1\right)^n B_n(x), && n \ge 0, \text{ and in particular for } n \ne 1,~B_n(0) = B_n(1)\\[3mu] E_n(1-x) &= \left(-1\right)^n E_n(x) \\[1ex] \left(-1\right)^n B_n(-x) &= B_n(x) + nx^{n-1} \\[3mu] \left(-1\right)^n E_n(-x) &= -E_n(x) + 2x^n \\[1ex] B_n\bigl(\tfrac12\bigr) &= \left(\frac{1}{2^{n-1}}-1\right) B_n, && n \geq 0\text{ from the multiplication theorems below.} \end{align} </math> [[Zhi-Wei Sun]] and Hao Pan <ref>{{cite journal |author1=Zhi-Wei Sun |author2=Hao Pan |journal=Acta Arithmetica |volume=125 |year=2006 |pages=21–39 |title=Identities concerning Bernoulli and Euler polynomials |issue=1 |arxiv=math/0409035 |doi=10.4064/aa125-1-3|bibcode=2006AcAri.125...21S |s2cid=10841415 }}</ref> established the following surprising symmetry relation: If {{math|1= ''r'' + ''s'' + ''t'' = ''n''}} and {{math|1= ''x'' + ''y'' + ''z'' = 1}}, then <math display="block">r[s,t;x,y]_n+s[t,r;y,z]_n+t[r,s;z,x]_n=0,</math> where <math display="block">[s,t;x,y]_n=\sum_{k=0}^n(-1)^k{s \choose k}{t\choose {n-k}} B_{n-k}(x)B_k(y).</math>

==Fourier series==

The [[Fourier series]] of the Bernoulli polynomials is also a [[Dirichlet series]], given by the expansion <math display="block">B_n(x) = -\frac{n!}{(2\pi i)^n}\sum_{k\not=0 }\frac{e^{2\pi ikx}}{k^n}= -2 n! \sum_{k=1}^{\infty} \frac{\cos\left(2 k \pi x- \frac{n \pi} 2 \right)}{(2 k \pi)^n}.</math> 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]] <math display="block">B_n(x) = -\Gamma(n+1) \sum_{k=1}^\infty \frac{ \exp (2\pi ikx) + e^{i\pi n} \exp (2\pi ik(1-x)) } { (2\pi ik)^n }. </math>

This expansion is valid only for {{math|0 ≤ ''x'' ≤ 1}} when {{math|''n'' ≥ 2}} and is valid for {{math|0 < ''x'' < 1}} when {{math|1=''n'' = 1}}.

The Fourier series of the Euler polynomials may also be calculated. Defining the functions <math display="block">\begin{align} C_\nu(x) &= \sum_{k=0}^\infty \frac {\cos((2k+1)\pi x)} {(2k+1)^\nu} \\[3mu] S_\nu(x) &= \sum_{k=0}^\infty \frac {\sin((2k+1)\pi x)} {(2k+1)^\nu} \end{align}</math> for <math>\nu > 1</math>, the Euler polynomial has the Fourier series <math display="block">\begin{align} C_{2n}(x) &= \frac{\left(-1\right)^n}{4(2n-1)!} \pi^{2n} E_{2n-1} (x) \\[1ex] S_{2n+1}(x) &= \frac{\left(-1\right)^n}{4(2n)!} \pi^{2n+1} E_{2n} (x). \end{align}</math> Note that the <math>C_\nu</math> and <math>S_\nu</math> are odd and even, respectively:<math display="block">\begin{align} C_\nu(x) &= -C_\nu(1-x) \\ S_\nu(x) &= S_\nu(1-x). \end{align}</math>

They are related to the [[Legendre chi function]] <math>\chi_\nu</math> as <math display="block">\begin{align} C_\nu(x) &= \operatorname{Re} \chi_\nu (e^{ix}) \\ S_\nu(x) &= \operatorname{Im} \chi_\nu (e^{ix}). \end{align}</math> ==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 [[#Representation by an integral operator|integral operators]], it follows that <math display="block">x^n = \frac {1}{n+1} \sum_{k=0}^n {n+1 \choose k} B_k (x)</math> and <math display="block">x^n = E_n (x) + \frac {1}{2} \sum_{k=0}^{n-1} {n \choose k} E_k (x).</math>

==Relation to falling factorial== The Bernoulli polynomials may be expanded in terms of the [[falling factorial]] <math>(x)_k</math> as <math display="block">B_{n+1}(x) = B_{n+1} + \sum_{k=0}^n \frac{n+1}{k+1} \left\{ \begin{matrix} n \\ k \end{matrix} \right\} (x)_{k+1} </math> where <math>B_n = B_n(0)</math> and <math display="block">\left\{ \begin{matrix} n \\ k \end{matrix} \right\} = S(n,k)</math> denotes the [[Stirling number of the second kind]]. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials: <math display="block">(x)_{n+1} = \sum_{k=0}^n \frac{n+1}{k+1} \left[ \begin{matrix} n \\ k \end{matrix} \right] \left(B_{k+1}(x) - B_{k+1} \right) </math> where <math display="block">\left[ \begin{matrix} n \\ k \end{matrix} \right] = s(n,k)</math> denotes the [[Stirling number of the first kind]].

==Multiplication theorems== The [[multiplication theorem]]s were given by [[Joseph Ludwig Raabe]] in 1851:

For a natural number {{math|''m''≥1}}, <math display="block">B_n(mx)= m^{n-1} \sum_{k=0}^{m-1} B_n{\left(x+\frac{k}{m}\right)}</math> <math display="block">\begin{align} E_n(mx) &= m^n \sum_{k=0}^{m-1} \left(-1\right)^k E_n{\left(x+\frac{k}{m}\right)} & \text{ for odd } m \\[1ex] E_n(mx) &= \frac{-2}{n+1} m^n \sum_{k=0}^{m-1} \left(-1\right)^k B_{n+1}{\left(x+\frac{k}{m}\right)} & \text{ for even } m \end{align}</math>

==Integrals== Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:<ref>{{cite journal |name-list-style=amp |author1=Takashi Agoh |author2=Karl Dilcher |journal=Journal of Mathematical Analysis and Applications |volume=381 |year=2011 |pages=10–16 |title=Integrals of products of Bernoulli polynomials | doi=10.1016/j.jmaa.2011.03.061 |doi-access=free }}</ref> *<math>\int_0^1 B_n(t) B_m(t)\,dt = (-1)^{n-1} \frac{m!\, n!}{(m+n)!} B_{n+m} \quad \text{for } m,n \geq 1 </math> *<math>\int_0^1 E_n(t) E_m(t)\,dt = (-1)^{n} 4 (2^{m+n+2}-1)\frac{m!\,n!}{(m+n+2)!} B_{n+m+2}</math>

Another integral formula states<ref>{{cite journal | author=Elaissaoui, Lahoucine | author2=Guennoun, Zine El Abidine | name-list-style=amp | title=Evaluation of log-tangent integrals by series involving ζ(2n+1)| journal=Integral Transforms and Special Functions | language=English | year=2017| volume=28 | issue=6 | pages=460–475 | doi=10.1080/10652469.2017.1312366 | arxiv=1611.01274 | s2cid=119132354 }}</ref> *<math>\int_0^{1}E_{n}\left( x +y\right)\log(\tan \frac{\pi}{2}x)\,dx= n! \sum_{k=1}^{\left\lfloor\frac {n+1}2\right\rfloor} \frac{(-1)^{k-1}}{ \pi^{2k}} \left( 2-2^{-2k} \right)\zeta(2k+1) \frac{y^ {n+1-2k}}{(n +1- 2k)!}</math> with the special case for <math>y=0</math> *<math>\int_0^{1}E_{2n-1}\left( x \right)\log(\tan \frac{\pi}{2}x)\,dx= \frac{(-1)^{n-1}(2n-1)!}{\pi^{2n}}\left( 2-2^{-2n} \right)\zeta(2n+1)</math> *<math>\int_0^{1}B_{2n-1}\left( x \right)\log(\tan \frac{\pi}{2}x)\,dx= \frac{(-1)^{n-1}}{\pi^{2n}}\frac{2^{2n-2}}{(2n-1)!}\sum_{k=1}^{n}( 2^{2k+1}-1 )\zeta(2k+1)\zeta(2n-2k)</math> *<math>\int_0^{1}E_{2n}\left( x \right)\log(\tan \frac{\pi}{2}x)\,dx=\int_0^{1}B_{2n}\left( x \right)\log(\tan \frac{\pi}{2}x)\,dx=0</math> *<math>\int_{0}^{1}{{{B}_{2n-1}}\left( x \right)\cot \left( \pi x \right)dx}=\frac{2\left( 2n-1 \right)!}{{{\left( -1 \right)}^{n-1}}{{\left( 2\pi \right)}^{2n-1}}}\zeta \left( 2n-1 \right)</math>

==Periodic Bernoulli polynomials== A '''periodic Bernoulli polynomial''' {{math|''P''''n''(''x'')}} is a Bernoulli polynomial evaluated at the [[fractional part]] of the argument {{math|''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 wave|sawtooth function]].

Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions.

The following properties are of interest, valid for all <math> x </math>: * <math>P_k(x)</math> is continuous for all <math> k > 1 </math> * <math>P_k'(x)</math> exists and is continuous for <math> k > 2 </math> * <math>P'_k(x) = k P_{k-1}(x)</math> for <math> k > 2 </math>

==See also== * [[Bernoulli numbers]] * [[Bernoulli polynomials of the second kind]] * [[Stirling polynomial]] * [[Polynomials calculating sums of powers of arithmetic progressions]]

==References== {{reflist}} {{refbegin}} * Milton Abramowitz and Irene A. Stegun, eds. ''[[Abramowitz and Stegun|Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables'', (1972) Dover, New York. ''(See Chapter 23)'' * {{Apostol IANT}} ''(See chapter 12.11)'' *{{dlmf|first=K. |last=Dilcher|id=24|title=Bernoulli and Euler Polynomials}} * {{Cite journal | last1 = Cvijović | first1 = Djurdje | last2 = Klinowski | first2 = Jacek | year = 1995 | title = New formulae for the Bernoulli and Euler polynomials at rational arguments | journal = [[Proceedings of the American Mathematical Society]] | volume = 123 | issue = 5 | pages = 1527–1535 | doi=10.1090/S0002-9939-1995-1283544-0 | doi-access = free | jstor = 2161144 }} *{{cite arXiv |first=Omran |last=Kouba |title=Lecture Notes, Bernoulli Polynomials and Applications |date=2016 |class=math.CA |eprint=1309.7560v2}} * {{Cite journal | doi = 10.1007/s11139-007-9102-0 | author1=Jesús Guillera | author1-link=Jesús Guillera | last2 = Sondow | first2 = Jonathan | year = 2008 | title = Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent | arxiv = math.NT/0506319 | journal = The Ramanujan Journal | volume = 16 | issue = 3| pages = 247–270 | s2cid = 14910435 }} ''(Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)'' * {{cite book | author=Hugh L. Montgomery | author-link=Hugh Montgomery (mathematician) |author2=Robert C. Vaughan |author-link2=Robert Charles Vaughan (mathematician) | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=978-0-521-84903-6 | pages=495–519 | publisher=Cambridge Univ. Press | location=Cambridge }} {{refend}}

==External links== * [https://dlmf.nist.gov/24.7 A list of integral identities involving Bernoulli polynomials] from [[NIST]]

{{authority control}} [[Category:Special functions]] [[Category:Number theory]] [[Category:Polynomials]]