Inmathematics, theBernoulli polynomials, named afterJacob Bernoulli, combine theBernoulli numbers andbinomial coefficients. They are used forseries expansion offunctions, and with theEuler–MacLaurin formula.

Thesepolynomials occur in the study of manyspecial functions and, in particular, theRiemann zeta function and theHurwitz zeta function. They are anAppell sequence (i.e. aSheffer sequence for the ordinaryderivative operator). For the Bernoulli polynomials, the number of crossings of thex-axis in theunit interval does not go up with thedegree. In the limit of large degree, they approach, when appropriately scaled, thesine and cosine functions.

A similar set of polynomials, based on a generating function, is the family ofEuler polynomials.

The Bernoulli polynomialsB_n can be defined by agenerating function. They also admit a variety of derived representations.

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

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

The Bernoulli polynomials are also given by$${\displaystyle B_n(x)=\frac{D}{e^D-1}{x}^n}$$ where${\displaystyle D\equiv \frac{\mathrm{d}}{\mathrm{d}x}}$  is differentiation with respect tox and the fraction is expanded as aformal power series. It follows that$${\displaystyle {\int }_a^x{B}_n(u)\mathrm{d}u=\frac{B_{n+1}(x)-B_{n+1}(a)}{n+1}.}$$ cf.§ Integrals below. By the same token, the Euler polynomials are given by$${\displaystyle E_n(x)=\frac{2}{e^D+1}{x}^n.}$$ 

The Bernoulli polynomials are also the unique polynomials determined by$${\displaystyle {\int }_x^{x+1}{B}_n(u)du=x^n.}$$ 

Theintegral transform$${\displaystyle (Tf)(x)={\int }_x^{x+1}f(u)du}$$ on polynomialsf, simply amounts to This can be used to produce theinversion formulae below.

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

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

That is similar to the series expression for theHurwitz zeta function in the complex plane. Indeed, there is the relationship$${\displaystyle B_n(x)=-n\zeta (1-n,x)}$$ where${\displaystyle \zeta (s,q)}$  is theHurwitz zeta function. The latter generalizes the Bernoulli polynomials, allowing for non-integer valuesofn.

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

This formula may be derived from an identity appearing above as follows. Since the forward difference operatorΔ equals$${\displaystyle \mathrm{\Delta }=e^D-1}$$ whereD is differentiation with respect tox, we have, from theMercator series,$${\displaystyle \frac{D}{e^D-1}=\frac{\log (\mathrm{\Delta }+1)}{\mathrm{\Delta }}=\sum _{n=0}^{\mathrm{\infty }}\frac{(-\mathrm{\Delta }{)}^n}{n+1}.}$$ 

As long as this operates on anmth-degree polynomial such as${\displaystyle x^m,}$  one may letn go from0 only uptom.

An integral representation for the Bernoulli polynomials is given by theNörlund–Rice integral, which follows from the expression as a finite difference.

An explicit formula for the Euler polynomials is given by$${\displaystyle E_n(x)=\sum _{k=0}^n\left[\frac{1}{2^k}\sum _{\ell =0}^n(-1{)}^{\ell }(\genfrac{}{}{0ex}{}{k}{\ell })(x+\ell {)}^n\right].}$$ 

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

Using either the aboveintegral representation of${\displaystyle x^n}$  or theidentity${\displaystyle B_n(x+1)-B_n(x)=n{x}^{n-1}}$ , we have$${\displaystyle \sum _{k=0}^x{k}^p={\int }_0^{x+1}{B}_p(t)dt=\frac{B_{p+1}(x+1)-B_{p+1}}{p+1}}$$ (assuming 0⁰ = 1).

The first few Bernoulli polynomials are: 

The first few Euler polynomials are: 

At highern the amount of variation in${\displaystyle B_n(x)}$  between${\displaystyle x=0}$  and${\displaystyle x=1}$  gets large. For instance,${\displaystyle B_{16}(0)=B_{16}(1)=}$ ${\displaystyle -\frac{3617}{510}\approx -7.09,}$  but${\displaystyle B_{16}(\frac{1}{2})=}$ ${\displaystyle \frac{118518239}{3342336}\approx \mathrm{7.09.}}$ Lehmer (1940)^[3] showed that the maximum value (M_n) of${\displaystyle B_n(x)}$  between0 and1 obeys unlessn is2 modulo 4, in which case$${\displaystyle M_n=\frac{2\zeta (n)n!}{(2\pi {)}^n}}$$ (where${\displaystyle \zeta (x)}$  is theRiemann zeta function), while the minimum (m_n) obeys$${\displaystyle m_n>\frac{-2n!}{(2\pi {)}^n}}$$ unlessn = 0 modulo 4 , in which case$${\displaystyle m_n=\frac{-2\zeta (n)n!}{(2\pi {)}^n}.}$$ 

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

The Bernoulli and Euler polynomials obey many relations fromumbral calculus: (Δ is theforward difference operator). Also,$${\displaystyle E_n(x+1)+E_n(x)=2x^n.}$$ Thesepolynomial sequences areAppell sequences: 

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

 Zhi-Wei Sun and Hao Pan^[4] established the following surprising symmetry relation: Ifr +s +t =n andx +y +z = 1, then$${\displaystyle r[s,t;x,y{]}_n+s[t,r;y,z{]}_n+t[r,s;z,x{]}_n=0,}$$ where$${\displaystyle [s,t;x,y{]}_n=\sum _{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{s}{k})(\genfrac{}{}{0ex}{}{t}{n-k})B_{n-k}(x)B_k(y).}$$ 

TheFourier series of the Bernoulli polynomials is also aDirichlet series, given by the expansion$${\displaystyle B_n(x)=-\frac{n!}{(2\pi i{)}^n}\sum _{k\ne 0}\frac{e^{2\pi ikx}}{k^n}=-2n!\sum _{k=1}^{\mathrm{\infty }}\frac{\cos \left(2k\pi x-\frac{n\pi }{2}\right)}{(2k\pi {)}^n}.}$$ Note the simple largen limit to suitably scaled trigonometric functions.

This is a special case of the analogous form for theHurwitz zeta function$${\displaystyle B_n(x)=-\mathrm{\Gamma }(n+1)\sum _{k=1}^{\mathrm{\infty }}\frac{\exp (2\pi ikx)+e^{i\pi n}\exp (2\pi ik(1-x))}{(2\pi ik{)}^n}.}$$ 

This expansion is valid only for0 ≤x ≤ 1 whenn ≥ 2 and is valid for0 <x < 1 whenn = 1.

The Fourier series of the Euler polynomials may also be calculated. Defining the functions for${\displaystyle \nu >1}$ , the Euler polynomial has the Fourier series Note that the${\displaystyle C_{\nu }}$  and${\displaystyle S_{\nu }}$  are odd and even, respectively: 

They are related to theLegendre chi function${\displaystyle {\chi }_{\nu }}$  as 

The Bernoulli and Euler polynomials may be inverted to express themonomial in terms of the polynomials.

Specifically, evidently from the above section onintegral operators, it follows that$${\displaystyle x^n=\frac{1}{n+1}\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n+1}{k})B_k(x)}$$ and$${\displaystyle x^n=E_n(x)+\frac{1}{2}\sum _{k=0}^{n-1}(\genfrac{}{}{0ex}{}{n}{k})E_k(x).}$$ 

The Bernoulli polynomials may be expanded in terms of thefalling factorial${\displaystyle (x{)}_k}$  as$${\displaystyle B_{n+1}(x)=B_{n+1}+\sum _{k=0}^n\frac{n+1}{k+1}\left\{\begin{array}{c}n\\ k\end{array}\right\}(x{)}_{k+1}}$$ where${\displaystyle B_n=B_n(0)}$  and$${\displaystyle \left\{\begin{array}{c}n\\ k\end{array}\right\}=S(n,k)}$$ denotes theStirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:$${\displaystyle (x{)}_{n+1}=\sum _{k=0}^n\frac{n+1}{k+1}\left[\begin{array}{c}n\\ k\end{array}\right]\left(B_{k+1}(x)-B_{k+1}\right)}$$ where$${\displaystyle \left[\begin{array}{c}n\\ k\end{array}\right]=s(n,k)}$$ denotes theStirling number of the first kind.

Themultiplication theorems were given byJoseph Ludwig Raabe in 1851:

For a natural numberm≥1,$${\displaystyle B_n(mx)=m^{n-1}\sum _{k=0}^{m-1}{B}_n\left(x+\frac{k}{m}\right)}$$  

Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:^[5]

• ${\displaystyle {\int }_0^1{B}_n(t)B_m(t)dt=(-1{)}^{n-1}\frac{m!n!}{(m+n)!}{B}_{n+m}\text{for }m,n\ge 1}$ 
• ${\displaystyle {\int }_0^1{E}_n(t)E_m(t)dt=(-1{)}^{n}4(2^{m+n+2}-1)\frac{m!n!}{(m+n+2)!}{B}_{n+m+2}}$ 

Another integral formula states^[6]

• ${\displaystyle {\int }_0^1{E}_n\left(x+y\right)\log (\tan \frac{\pi }{2}x)dx=n!\sum _{k=1}^{\lfloor \frac{n+1}{2}\rfloor }\frac{(-1{)}^{k-1}}{{\pi }^{2k}}\left(2-2^{-2k}\right)\zeta (2k+1)\frac{y^{n+1-2k}}{(n+1-2k)!}}$ 

with the special case for${\displaystyle y=0}$ 

• ${\displaystyle {\int }_0^1{E}_{2n-1}\left(x\right)\log (\tan \frac{\pi }{2}x)dx=\frac{(-1{)}^{n-1}(2n-1)!}{{\pi }^{2n}}\left(2-2^{-2n}\right)\zeta (2n+1)}$ 
• ${\displaystyle {\int }_0^1{B}_{2n-1}\left(x\right)\log (\tan \frac{\pi }{2}x)dx=\frac{(-1{)}^{n-1}}{{\pi }^{2n}}\frac{2^{2n-2}}{(2n-1)!}\sum _{k=1}^n(2^{2k+1}-1)\zeta (2k+1)\zeta (2n-2k)}$ 
• ${\displaystyle {\int }_0^1{E}_{2n}\left(x\right)\log (\tan \frac{\pi }{2}x)dx={\int }_0^1{B}_{2n}\left(x\right)\log (\tan \frac{\pi }{2}x)dx=0}$ 
• ${\displaystyle {\int }_0^1{B}_{2n-1}\left(x\right)\cot \left(\pi x\right)dx=\frac{2\left(2n-1\right)!}{{\left(-1\right)}^{n-1}{\left(2\pi \right)}^{2n-1}}\zeta \left(2n-1\right)}$ 

Aperiodic Bernoulli polynomialP_n(x) is a Bernoulli polynomial evaluated at thefractional part of the argumentx. These functions are used to provide theremainder term in theEuler–Maclaurin formula relating sums to integrals. The first polynomial is asawtooth function.

Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions, andP₀(x) is not even a function, being the derivative of a sawtooth and so aDirac comb.

The following properties are of interest, valid for all${\displaystyle x}$ :

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

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

• A list of integral identities involving Bernoulli polynomials fromNIST