Bernoulli polynomials

File:Bernoulli polynomials.svg

Bernoulli polynomials

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 <math display="block">\frac{t e^{xt}}{e^t-1}= \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}.</math> The 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 numbers, 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 Template:Mvar 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. Template:Slink 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 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 Template:Nobr

The inner sum may be understood to be the Template:Mvarth 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 Template:Math equals <math display="block">\Delta = e^D - 1</math> where Template:Mvar is differentiation with respect to Template:Mvar, 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 Template:Mvarth-degree polynomial such as <math>x^m,</math> one may let Template:Mvar go from Template:Math only up Template:Nobr

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 pth powers

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Using either the above integral representation of <math>x^n</math> or the 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 Template:Mvar 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> Template:Nobr showed that the maximum value (Template:Mvar) of <math>B_n(x)</math> between Template:Math and Template:Math obeys <math display="block">M_n < \frac{2n!}{(2\pi)^n}</math> unless Template:Mvar is Template:Nobr 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 (Template:Mvar) obeys <math display="block">m_n > \frac{ -2 n!}{(2\pi)^n}</math> unless Template:Nobr 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> (Template:Math is the forward difference operator). Also, <math display="block"> E_n(x+1) + E_n(x) = 2x^n.</math> These polynomial sequences are Appell sequences: <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 sequences. (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>Template:Cite journal</ref> established the following surprising symmetry relation: If Template:Math and Template:Math, 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 Template:Math when Template:Math and is valid for Template:Math when Template:Math.

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 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 theorems were given by Joseph Ludwig Raabe in 1851:

For a natural number Template:Math, <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>Template:Cite journal</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>Template:Cite journal</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 Template:Math is a Bernoulli polynomial evaluated at the fractional part of the argument Template:Math. 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 Template:Math 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 <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

• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (1972) Dover, New York. (See Chapter 23)
• Template:Apostol IANT (See chapter 12.11)
• Template:Dlmf
• Template:Cite journal
• Template:Cite journal (Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)
• Template:Cite book

External links

• A list of integral identities involving Bernoulli polynomials from NIST