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 Bn 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
The generating function for the Euler polynomials is
Explicit formula
for n ≥ 0, where Bk are the Bernoulli numbers, and Ek are the Euler numbers.
Representation by a differential operator
The Bernoulli polynomials are also given by
where D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series. It follows that
cf. integrals 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.
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 ζ(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 xm; that is,
where Δ 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 Δ equals
where D is differentiation with respect to x, we have, from the Mercator series,
As long as this operates on an mth-degree polynomial such as xm, 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
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).
The Bernoulli and Euler numbers
The Bernoulli numbers are given by
This definition gives for .
An alternate convention defines the Bernoulli numbers as
The two conventions differ only for since .
The Euler numbers are given by
Explicit expressions for low degrees
The first few Bernoulli polynomials are:
The first few Euler polynomials are:
Maximum and minimum
At higher n, the amount of variation in Bn(x) between x = 0 and x = 1 gets large. For instance,
which shows that the value at x = 0 (and at x = 1) is −3617/510 ≈ −7.09, while at x = 1/2, the value is 118518239/3342336 ≈ +7.09. D.H. Lehmer showed that the maximum value of Bn(x) between 0 and 1 obeys
unless n is 2 modulo 4, in which case
(where is the Riemann zeta function), while the minimum obeys
unless n is 0 modulo 4, 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:
(Δ 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 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
and
for , the Euler polynomial has the Fourier series
and
Note that the and are odd and even, respectively:
and
They are related to the Legendre chi function as
and
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, and 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 :
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
Special functions
Number theory
Polynomials