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 Bn 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
\frac{t e^{xt}}{e^t-1}= \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}.
The exponential generating function for the Euler polynomials is
\frac{2 e^{xt}}{e^t+1}= \sum_{n=0}^\infty E_n(x) \frac{t^n}{n!}.
Explicit formula
B_n(x) = \sum_{k=0}^n {n \choose k} B_{n-k} x^k,
E_m(x)=
\sum_{k=0}^m {m \choose k} \frac{E_k}{2^k}
\left(x-\tfrac12\right)^{m-k} .
for n \geq 0, where B_k are the Bernoulli numbers, and E_k are the Euler numbers. It follows that B_n(0) = B_n and E_m\big(\tfrac{1}{2}\big) = \tfrac{1}{2^m} E_m.
Representation by a differential operator
The Bernoulli polynomials are also given by
\ B_n(x) = \frac{ D }{\ e^D -1\ }\ x^n\
where \ D \equiv \frac{ \mathrm{d} }{\ \mathrm{d} x\ }\ is differentiation with respect to x and the fraction is expanded as a formal power series. It follows that
\ \int_a^x\ B_n(u)\ \mathrm{d}\ u = \frac{\ B_{n+1}(x) - B_{n+1}(a)\ }{ n + 1 } ~.
cf. § Integrals below. By the same token, the Euler polynomials are given by
\ E_n(x) = \frac{ 2 }{\ e^D + 1\ }\ x^n ~.
Representation by an integral operator
The Bernoulli polynomials are also the unique polynomials determined by
\int_x^{x+1} B_n(u)\,du = x^n.
(Tf)(x) = \int_x^{x+1} f(u)\,du
on polynomials f, simply amounts to
\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}
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_{m}(x)=m \int_{0}^{x} B_{m-1}(t)\,dt-m\int_{0}^{1} \int_0^t B_{m-1}(s)\,ds dt.
Another explicit formula
An explicit formula for the Bernoulli polynomials is given by
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].
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 \zeta(1 - n,\,x)
where \zeta(s,\,q) 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 x^m, that is,
\Delta^n x^m = \sum_{k=0}^n (-1)^{n - k}{n \choose k}(x + k)^m
where \Delta is the forward difference operator. Thus, one may write
B_n(x) = \sum_{k=0}^n \frac{(-1)^k}{k + 1}\Delta^k x^n.
This formula may be derived from an identity appearing above as follows. Since the forward difference operator Δ equals
\Delta = e^D - 1
where D is differentiation with respect to x, we have, from the Mercator series,
\frac{ D }{e^D - 1} = \frac{\log(\Delta + 1)}{\Delta} = \sum_{n=0}^\infty \frac{(-\Delta)^n }{n + 1}.
As long as this operates on an mth-degree polynomial such as x^m, 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
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] .
The above follows analogously, using the fact that
\frac{2}{e^D + 1} = \frac{1}{1 + \tfrac12 \Delta} = \sum_{n = 0}^\infty \bigl( {-\tfrac{1}{2}} \Delta \bigr)^n .
Sums of pth powers
Using either the above integral representation of x^n or the identity B_n(x + 1) - B_n(x) = nx^{n-1}, we have
\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}
(assuming 00 = 1).
Explicit expressions for low degrees
The first few Bernoulli polynomials are:
\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}
The first few Euler polynomials are:
\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}
Maximum and minimum
At higher n 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} \approx -7.09, but B_{16}\bigl(\tfrac12\bigr) = {}\tfrac{118518239}{3342336} \approx 7.09. Lehmer (1940)[3] showed that the maximum value (Mn) of B_n(x) between 0 and 1 obeys
M_n < \frac{2n!}{(2\pi)^n}
unless n is 2 modulo 4, in which case
M_n = \frac{2\zeta (n)\,n!}{(2\pi)^n}
(where \zeta(x) is the Riemann zeta function), while the minimum (mn) obeys
m_n > \frac{ -2 n!}{(2\pi)^n}
unless n = 0 modulo 4 , in which case
m_n = \frac{-2 \zeta(n)\,n! }{(2\pi)^n}.
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:
\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}
(Δ is the forward difference operator). Also,
E_n(x+1) + E_n(x) = 2x^n.
These polynomial sequences are Appell sequences:
\begin{align}
B_n'(x) &= n B_{n-1}(x), \\[3mu]
E_n'(x) &= n E_{n-1}(x).
\end{align}
Translations
\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}
These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials are another example.)
Symmetries
\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}
Zhi-Wei Sun and Hao Pan [4] established the following surprising symmetry relation: If r + s + t = n and x + y + z = 1, 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=\sum_{k=0}^n(-1)^k{s \choose k}{t\choose {n-k}} B_{n-k}(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) = -\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}.
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) = -\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 }.
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
\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}
for \nu > 1, the Euler polynomial has the Fourier series
\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}
Note that the C_\nu and S_\nu are odd and even, respectively:
\begin{align}
C_\nu(x) &= -C_\nu(1-x) \\
S_\nu(x) &= S_\nu(1-x).
\end{align}
They are related to the Legendre chi function \chi_\nu as
\begin{align}
C_\nu(x) &= \operatorname{Re} \chi_\nu (e^{ix}) \\
S_\nu(x) &= \operatorname{Im} \chi_\nu (e^{ix}).
\end{align}
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 = \frac {1}{n+1} \sum_{k=0}^n {n+1 \choose k} B_k (x)
and
x^n = E_n (x) + \frac {1}{2} \sum_{k=0}^{n-1} {n \choose k} E_k (x).
Relation to falling factorial
The Bernoulli polynomials may be expanded in terms of the falling factorial (x)_k as
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}
where B_n = B_n(0) and
\left\{ \begin{matrix} n \\ k \end{matrix} \right\} = 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)_{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)
where
\left[ \begin{matrix} n \\ k \end{matrix} \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 m≥1,
B_n(mx)= m^{n-1} \sum_{k=0}^{m-1} B_n{\left(x+\frac{k}{m}\right)}
\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}
Integrals
Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:[5]
\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\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}
Another integral formula states[6]
\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)!}
with the special case for y=0
\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)\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)\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\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)
Periodic Bernoulli polynomials
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.
The following properties are of interest, valid for all x:
P_k(x)is continuous for allk > 1P_k'(x)exists and is continuous fork > 2P'_k(x) = k P_{k-1}(x)fork > 2
See also
- Bernoulli numbers
- Bernoulli polynomials of the second kind
- Stirling polynomial
- Polynomials calculating sums of powers of arithmetic progressions
References
- ^ 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/
- ^ Lehmer, D.H. (1940). "On the maxima and minima of Bernoulli polynomials". American Mathematical Monthly. 47 (8): 533–538. doi:10.1080/00029890.1940.11991015
- ^ 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
- ^ 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)
- (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
- Kouba, Omran (2016). "Lecture Notes, Bernoulli Polynomials and Applications". arXiv:1309.7560v2
- Jesús Guillera & 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. Vol. 97. Cambridge tracts in advanced mathematics. Cambridge: Cambridge Univ. Press. pp. 495–519. ISBN 978-0-521-84903-6.