Bernoulli polynomials

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 $$\frac{t{e}^{xt}}{e^t-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_n(x)\frac{t^n}{n!}.$$ The generating function for the Euler polynomials is $$\frac{2e^{xt}}{e^t+1}=\sum _{n=0}^{\mathrm{\infty }}{E}_n(x)\frac{t^n}{n!}.$$

Explicit formula

$$B_n(x)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})B_{n-k}{x}^k,$$ $$E_m(x)=\sum _{k=0}^m(\genfrac{}{}{0ex}{}{m}{k})\frac{E_k}{2^k}{(x-\frac{1}{2})}^{m-k}.$$ for $n\ge 0$, where $B_k$ are the Bernoulli numbers, and $E_k$ are the Euler numbers. It follows that $B_n(0)=B_n$ and $E_m\left(\frac{1}{2}\right)=\frac{1}{2^m}{E}_m$.

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 Template:Mvar and the fraction is expanded as a formal power series. It follows that $$\underset{a}{\overset{x}{\int }}{B}_n(u)\mathrm{d}u=\frac{B_{n+1}(x)-B_{n+1}(a)}{n+1}.$$ cf. Template:Slink 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 $$\underset{x}{\overset{x+1}{\int }}{B}_n(u)du=x^n.$$

The integral transform $$(Tf)(x)=\underset{x}{\overset{x+1}{\int }}f(u)du$$ on polynomials f, simply amounts to $$\begin{array}{cc}(Tf)(x)=\frac{e^D-1}{D}f(x)& =\sum _{n=0}^{\mathrm{\infty }}\frac{D^n}{(n+1)!}f(x)\\ & =f(x)+\frac{f^{\prime }(x)}{2}+\frac{f^{″}(x)}{6}+\frac{f^{‴}(x)}{24}+\cdots .\end{array}$$ This can be used to produce the inversion formulae below.

Integral Recurrence

In, it is deduced and proved that the Bernoulli polynomials can be obtained by the following integral recurrence $$B_m(x)=m\underset{0}{\overset{x}{\int }}{B}_{m-1}(t)dt-m\underset{0}{\overset{1}{\int }}\underset{0}{\overset{t}{\int }}{B}_{m-1}(s)dsdt.$$

Another explicit formula

An explicit formula for the Bernoulli polynomials is given by $$B_n(x)=\sum _{k=0}^n[\frac{1}{k+1}\sum _{\ell =0}^k(-1{)}^{\ell }(\genfrac{}{}{0ex}{}{k}{\ell })(x+\ell {)}^n].$$

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 Template:Nobr

The inner sum may be understood to be the Template:Mvarth forward difference of $x^m,$ that is, $${\mathrm{\Delta }}^n{x}^m=\sum _{k=0}^n(-1{)}^{n-k}(\genfrac{}{}{0ex}{}{n}{k})(x+k{)}^m$$ where $\mathrm{\Delta }$ is the forward difference operator. Thus, one may write $$B_n(x)=\sum _{k=0}^n\frac{(-1{)}^k}{k+1}{\mathrm{\Delta }}^k{x}^n.$$

This formula may be derived from an identity appearing above as follows. Since the forward difference operator Template:Math equals $$\mathrm{\Delta }=e^D-1$$ where Template:Mvar is differentiation with respect to Template:Mvar, we have, from the Mercator series, $$\frac{D}{e^D-1}=\frac{\log (\mathrm{\Delta }+1)}{\mathrm{\Delta }}=\sum _{n=0}^{\mathrm{\infty }}\frac{(-\mathrm{\Delta }{)}^n}{n+1}.$$

As long as this operates on an Template:Mvarth-degree polynomial such as $x^m,$ 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 $$E_n(x)=\sum _{k=0}^n[\frac{1}{2^k}\sum _{\ell =0}^n(-1{)}^{\ell }(\genfrac{}{}{0ex}{}{k}{\ell })(x+\ell {)}^n].$$

The above follows analogously, using the fact that $$\frac{2}{e^D+1}=\frac{1}{1+\frac{1}{2}\mathrm{\Delta }}=\sum _{n=0}^{\mathrm{\infty }}(-\frac{1}{2}\mathrm{\Delta }{)}^n.$$

Sums of pth powers

Using either the above integral representation of $x^n$ or the identity $B_n(x+1)-B_n(x)=n{x}^{n-1}$, we have $$\sum _{k=0}^x{k}^p=\underset{0}{\overset{x+1}{\int }}{B}_p(t)dt=\frac{B_{p+1}(x+1)-B_{p+1}}{p+1}$$ (assuming 0⁰ = 1).

Explicit expressions for low degrees

The first few Bernoulli polynomials are: $$\begin{array}{cccc}{B}_0(x)& =1,& B_4(x)& =x^4-2x^3+x^2-\frac{1}{30},\\ B_1(x)& =x-\frac{1}{2},& B_5(x)& =x^5-\frac{5}{2}{x}^4+\frac{5}{3}{x}^3-\frac{1}{6}x,\\ B_2(x)& =x^2-x+\frac{1}{6},& B_6(x)& =x^6-3x^5+\frac{5}{2}{x}^4-\frac{1}{2}{x}^2+\frac{1}{42},\\ B_3(x)& =x^3-\frac{3}{2}{x}^2+\frac{1}{2}x\phantom{|},& & ⋮\end{array}$$

The first few Euler polynomials are: $$\begin{array}{cccc}{E}_0(x)& =1,& E_4(x)& =x^4-2x^3+x,\\ E_1(x)& =x-\frac{1}{2},& E_5(x)& =x^5-\frac{5}{2}{x}^4+\frac{5}{2}{x}^2-\frac{1}{2},\\ E_2(x)& =x^2-x,& E_6(x)& =x^6-3x^5+5x^3-3x,\\ E_3(x)& =x^3-\frac{3}{2}{x}^2+\frac{1}{4},& & ⋮\end{array}$$

Maximum and minimum

At higher Template:Mvar the amount of variation in $B_n(x)$ between $x=0$ and $x=1$ gets large. For instance, $B_{16}(0)=B_{16}(1)=$$-\frac{3617}{510}\approx -7.09,$ but $B_{16}\left(\frac{1}{2}\right)=$$\frac{118518239}{3342336}\approx 7.09.$ Template:Nobr showed that the maximum value (Template:Mvar) of $B_n(x)$ between Template:Math and Template:Math obeys $$M_n<\frac{2n!}{(2\pi {)}^n}$$ unless Template:Mvar is Template:Nobr in which case $$M_n=\frac{2\zeta (n)n!}{(2\pi {)}^n}$$ (where $\zeta (x)$ is the Riemann zeta function), while the minimum (Template:Mvar) obeys $$m_n>\frac{-2n!}{(2\pi {)}^n}$$ unless Template:Nobr 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: $$\begin{array}{cc}\mathrm{\Delta }{B}_n(x)& =B_n(x+1)-B_n(x)=n{x}^{n-1},\\ \mathrm{\Delta }{E}_n(x)& =E_n(x+1)-E_n(x)=2(x^n-E_n(x)).\end{array}$$ (Template:Math is the forward difference operator). Also, $$E_n(x+1)+E_n(x)=2x^n.$$ These polynomial sequences are Appell sequences: $$\begin{array}{cc}{B_n}^{\prime }(x)& =n{B}_{n-1}(x),\\ {E_n}^{\prime }(x)& =n{E}_{n-1}(x).\end{array}$$

Translations

$$\begin{array}{cc}{B}_n(x+y)& =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})B_k(x)y^{n-k}\\ E_n(x+y)& =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})E_k(x)y^{n-k}\end{array}$$ These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials are another example.)

Symmetries

$$\begin{array}{cccc}{B}_n(1-x)& ={(-1)}^n{B}_n(x),& & n\ge 0,\text{ and in particular for }n\ne 1,B_n(0)=B_n(1)\\ E_n(1-x)& ={(-1)}^n{E}_n(x)\\ {(-1)}^n{B}_n(-x)& =B_n(x)+n{x}^{n-1}\\ {(-1)}^n{E}_n(-x)& =-E_n(x)+2x^n\\ B_n\left(\frac{1}{2}\right)& =(\frac{1}{2^{n-1}}-1)B_n,& & n\ge 0\text{ from the multiplication theorems below.}\end{array}$$ Zhi-Wei Sun and Hao Pan established the following surprising symmetry relation: If Template:Math and Template:Math, then $$r[s,t;x,y{]}_n+s[t,r;y,z{]}_n+t[r,s;z,x{]}_n=0,$$ where $$[s,t;x,y{]}_n=\sum _{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{s}{k})(\genfrac{}{}{0ex}{}{t}{n-k})B_{n-k}(x)B_k(y).$$

Fourier series

The Fourier series of the Bernoulli polynomials is also a Dirichlet series, given by the expansion $$B_n(x)=-\frac{n!}{(2\pi i{)}^n}\sum _{k=0}\frac{e^{2\pi ikx}}{k^n}=-2n!\sum _{k=1}^{\mathrm{\infty }}\frac{\cos (2k\pi x-\frac{n\pi }{2})}{(2k\pi {)}^n}.$$ 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 $$B_n(x)=-\mathrm{\Gamma }(n+1)\sum _{k=1}^{\mathrm{\infty }}\frac{\exp (2\pi ikx)+e^{i\pi n}\exp (2\pi ik(1-x))}{(2\pi ik{)}^n}.$$

This expansion is valid only for 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 $$\begin{array}{cc}{C}_{\nu }(x)& =\sum _{k=0}^{\mathrm{\infty }}\frac{\cos ((2k+1)\pi x)}{(2k+1{)}^{\nu }}\\ S_{\nu }(x)& =\sum _{k=0}^{\mathrm{\infty }}\frac{\sin ((2k+1)\pi x)}{(2k+1{)}^{\nu }}\end{array}$$ for $\nu >1$, the Euler polynomial has the Fourier series $$\begin{array}{cc}{C}_{2n}(x)& =\frac{{(-1)}^n}{4(2n-1)!}{\pi }^{2n}{E}_{2n-1}(x)\\ S_{2n+1}(x)& =\frac{{(-1)}^n}{4(2n)!}{\pi }^{2n+1}{E}_{2n}(x).\end{array}$$ Note that the $C_{\nu }$ and $S_{\nu }$ are odd and even, respectively:$$\begin{array}{cc}{C}_{\nu }(x)& =-C_{\nu }(1-x)\\ S_{\nu }(x)& =S_{\nu }(1-x).\end{array}$$

They are related to the Legendre chi function ${\chi }_{\nu }$ as $$\begin{array}{cc}{C}_{\nu }(x)& =\mathrm{Re}{\chi }_{\nu }(e^{ix})\\ S_{\nu }(x)& =\mathrm{Im}{\chi }_{\nu }(e^{ix}).\end{array}$$

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 $$x^n=\frac{1}{n+1}\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n+1}{k})B_k(x)$$ and $$x^n=E_n(x)+\frac{1}{2}\sum _{k=0}^{n-1}(\genfrac{}{}{0ex}{}{n}{k})E_k(x).$$

Relation to falling factorial

The Bernoulli polynomials may be expanded in terms of the falling factorial $(x{)}_k$ as $$B_{n+1}(x)=B_{n+1}+\sum _{k=0}^n\frac{n+1}{k+1}\left\{\begin{array}{c}n\\ k\end{array}\right\}(x{)}_{k+1}$$ where $B_n=B_n(0)$ and $$\left\{\begin{array}{c}n\\ k\end{array}\right\}=S(n,k)$$ denotes the Stirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials: $$(x{)}_{n+1}=\sum _{k=0}^n\frac{n+1}{k+1}\left[\begin{array}{c}n\\ k\end{array}\right](B_{k+1}(x)-B_{k+1})$$ where $$\left[\begin{array}{c}n\\ k\end{array}\right]=s(n,k)$$ 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 Template:Math, $$B_n(mx)=m^{n-1}\sum _{k=0}^{m-1}{B}_n(x+\frac{k}{m})$$ $$\begin{array}{ccc}{E}_n(mx)& =m^n\sum _{k=0}^{m-1}{(-1)}^k{E}_n(x+\frac{k}{m})& \text{ for odd }m\\ E_n(mx)& =\frac{-2}{n+1}{m}^n\sum _{k=0}^{m-1}{(-1)}^k{B}_{n+1}(x+\frac{k}{m})& \text{ for even }m\end{array}$$

Integrals

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

• $\underset{0}{\overset{1}{\int }}{B}_n(t)B_m(t)dt=(-1{)}^{n-1}\frac{m!n!}{(m+n)!}{B}_{n+m}\text{for }m,n\ge 1$
• $\underset{0}{\overset{1}{\int }}{E}_n(t)E_m(t)dt=(-1{)}^{n}4(2^{m+n+2}-1)\frac{m!n!}{(m+n+2)!}{B}_{n+m+2}$

Another integral formula states

• $\underset{0}{\overset{1}{\int }}{E}_n(x+y)\log (\tan \frac{\pi }{2}x)dx=n!\sum _{k=1}^{\lfloor \frac{n+1}{2}\rfloor }\frac{(-1{)}^{k-1}}{{\pi }^{2k}}(2-2^{-2k})\zeta (2k+1)\frac{y^{n+1-2k}}{(n+1-2k)!}$

with the special case for $y=0$

• $\underset{0}{\overset{1}{\int }}{E}_{2n-1}\left(x\right)\log (\tan \frac{\pi }{2}x)dx=\frac{(-1{)}^{n-1}(2n-1)!}{{\pi }^{2n}}(2-2^{-2n})\zeta (2n+1)$
• $\underset{0}{\overset{1}{\int }}{B}_{2n-1}\left(x\right)\log (\tan \frac{\pi }{2}x)dx=\frac{(-1{)}^{n-1}}{{\pi }^{2n}}\frac{2^{2n-2}}{(2n-1)!}\sum _{k=1}^n(2^{2k+1}-1)\zeta (2k+1)\zeta (2n-2k)$
• $\underset{0}{\overset{1}{\int }}{E}_{2n}\left(x\right)\log (\tan \frac{\pi }{2}x)dx=\underset{0}{\overset{1}{\int }}{B}_{2n}\left(x\right)\log (\tan \frac{\pi }{2}x)dx=0$
• $\underset{0}{\overset{1}{\int }}{B}_{2n-1}\left(x\right)\cot \left(\pi x\right)dx=\frac{2(2n-1)!}{{(-1)}^{n-1}{\left(2\pi \right)}^{2n-1}}\zeta (2n-1)$

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 $x$:

• $P_k(x)$ is continuous for all $k>1$
• ${P_k}^{\prime }(x)$ exists and is continuous for $k>2$
• $P{\text{'}}_k(x)=k{P}_{k-1}(x)$ 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)
• 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