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
The exponential generating function for the Euler polynomials is
Explicit formula
for , where are the Bernoulli numbers, and are the Euler numbers. It follows that and .
Representation by a differential operator
The Bernoulli polynomials are also given by
where is differentiation with respect to and the fraction is expanded as a formal power series. It follows that
cf. below. By the same token, the Euler polynomials are given by
Representation by an integral operator
The Bernoulli polynomials are also the unique polynomials determined by
The integral transform
on polynomials f, simply amounts to
This can be used to produce the inversion formulae below.
Integral recurrence
In,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/11232/174Sergio 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/ it is deduced and proved that the Bernoulli polynomials can be obtained by the following integral recurrence
Another explicit formula
An explicit formula for the Bernoulli polynomials is given by
That is similar to the series expression for the Hurwitz zeta function in the complex plane. Indeed, there is the relationship
where is the Hurwitz zeta function. The latter generalizes the Bernoulli polynomials, allowing for non-integer values
The inner sum may be understood to be the th forward difference of that is,
where is the forward difference operator. Thus, one may write
This formula may be derived from an identity appearing above as follows. Since the forward difference operator equals
where is differentiation with respect to , we have, from the Mercator series,
As long as this operates on an th-degree polynomial such as one may let go from only up
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
The above follows analogously, using the fact that
Sums of pth powers
Using either the above integral representation of or the identity , we have
(assuming 00 = 1).
Explicit expressions for low degrees The first few Bernoulli polynomials are:
The first few Euler polynomials are:
Maximum and minimum
At higher the amount of variation in between and gets large. For instance, but showed that the maximum value () of between and obeys
unless is in which case
(where is the Riemann zeta function), while the minimum () obeys
unless in which case
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:
( is the forward difference operator). Also,
These polynomial sequences are Appell sequences:
Translations
These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials are another example.)
Symmetries
Zhi-Wei Sun and Hao Pan established the following surprising symmetry relation: If and , then
where
Fourier series
The Fourier series of the Bernoulli polynomials is also a Dirichlet series, given by the expansion
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
This expansion is valid only for when and is valid for when .
The Fourier series of the Euler polynomials may also be calculated. Defining the functions
for , the Euler polynomial has the Fourier series
Note that the and are odd and even, respectively:
They are related to the Legendre chi function as
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
and
Relation to falling factorial The Bernoulli polynomials may be expanded in terms of the falling factorial as
where and
denotes the Stirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:
where
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 ,
Integrals Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:
Another integral formula states
with the special case for
Periodic Bernoulli polynomials A periodic Bernoulli polynomial is a Bernoulli polynomial evaluated at the fractional part of the argument . 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 : is continuous for all exists and is continuous for for
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) (See chapter 12.11)
(Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)
External links A list of integral identities involving Bernoulli polynomials from NIST