Bernoulli polynomials
Encyclopedia
In mathematics
, the Bernoulli polynomials occur in the study of many special functions
and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative
operator. Unlike orthogonal polynomials
, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval
does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions
.
. Which among them should be taken to be the definition may depend on one's purposes.
for n ≥ 0, where bk are the Bernoulli number
s.
for the Bernoulli polynomials is
The generating function for the Euler polynomials is
where D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series
.
The integral operator
on polynomials f, is the same as
Note the remarkable similarity to the globally convergent series expression for the Hurwitz zeta function. Indeed, one has
where ζ(s, q) is the Hurwitz zeta; thus, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to 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 is equal to
where D is differentiation with respect to x, we have
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
This may also be written in terms of the Euler numbers Ek as
See Faulhaber's formula for more on this.
s are given by
An alternate convention defines the Bernoulli numbers as
. This definition gives Bn = −nζ(1 − n) where for n = 0 and n = 1 the expression −nζ(1 − n) is to be understood as
limx → n −xζ(1 − x).
The two conventions differ only for n = 1 since B1(1) = 1/2 = −B1(0).
The Euler numbers are given by
The first few Euler polynomials are
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.
:
(Δ is the forward difference operator).
These polynomial sequence
s are Appell sequences:
These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials
are another example.)
Zhi-Wei Sun
and Hao Pan established the following surprising symmetric relation: If r + s + t = n and x + y + z = 1, then
where
of the Bernoulli polynomials is also a Dirichlet series, given by the expansion
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
in terms of the polynomials. Specifically, one has
and
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.
s were given by Joseph Ludwig Raabe
in 1851:
Definite integrals
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Bernoulli polynomials occur in the study of many special functions
Special functions
Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications....
and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
operator. Unlike orthogonal polynomials
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
.
Representations
The Bernoulli polynomials Bn admit a variety of different representationsRepresentation (mathematics)
In mathematics, representation is a very general relationship that expresses similarities between objects. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the...
. Which among them should be taken to be the definition may depend on one's purposes.
Explicit formula
for n ≥ 0, where bk are the Bernoulli number
Bernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
s.
Generating functions
The generating functionGenerating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
for the Bernoulli polynomials is
The generating function for the Euler polynomials is
Representation by a differential operator
The Bernoulli polynomials are also given bywhere D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
.
Representation by an integral operator
The Bernoulli polynomials are the unique polynomials determined byThe integral operator
on polynomials f, is the same as
Another explicit formula
An explicit formula for the Bernoulli polynomials is given byNote the remarkable similarity to the globally convergent series expression for the Hurwitz zeta function. Indeed, one has
where ζ(s, q) is the Hurwitz zeta; thus, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to 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 is equal to
where D is differentiation with respect to x, we have
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
This may also be written in terms of the Euler numbers Ek as
Sums of pth powers
We haveSee Faulhaber's formula for more on this.
The Bernoulli and Euler numbers
The Bernoulli numberBernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
s are given by
An alternate convention defines the Bernoulli numbers as
. This definition gives Bn = −nζ(1 − n) where for n = 0 and n = 1 the expression −nζ(1 − n) is to be understood aslimx → n −xζ(1 − x).
The two conventions differ only for n = 1 since B1(1) = 1/2 = −B1(0).
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 obeysunless 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 calculusUmbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to 'prove' them. These techniques were introduced by and are sometimes called Blissard's symbolic method...
:
(Δ is the forward difference operator).
These polynomial sequence
Polynomial sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial...
s are Appell sequences:
Translations
These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where...
are another example.)
Symmetries
Zhi-Wei Sun
Zhi-Wei Sun
Zhi-Wei Sun is a Chinese mathematician, working primarily on number theory, combinatorics, and group theory. Currently he works as a professor in Nanjing University....
and Hao Pan established the following surprising symmetric relation: If r + s + t = n and x + y + z = 1, then
where
Fourier series
The Fourier seriesFourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
of the Bernoulli polynomials is also a Dirichlet series, given by the expansion
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 seriesand
Note that the
and are odd and even, respectively:and
They are related to the Legendre chi function
asand
Inversion
The Bernoulli and Euler polynomials may be inverted to express the monomialMonomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
in terms of the polynomials. Specifically, one has
and
Relation to falling factorial
The Bernoulli polynomials may be expanded in terms of the falling factorial aswhere
anddenotes 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 theoremMultiplication theorem
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name...
s were given by Joseph Ludwig Raabe
Joseph Ludwig Raabe
Joseph Ludwig Raabe was a Swiss mathematician.-Life:As his parents were quite poor, Raabe was forced to earn his living from a very early age by giving private lessons. He began to study mathematics in 1820 at the Polytechnicum in Vienna, Austria...
in 1851:
Integrals
Indefinite integralsDefinite integrals
Periodic Bernoulli polynomials
A periodic Bernoulli polynomial Pn(x) is a Bernoulli polynomial evaluated at the fractional partFractional part
All real numbers can be written in the form n + r where n is an integer and the remaining fractional part r is a nonnegative real number less than one...
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
Sawtooth wave
The sawtooth wave is a kind of non-sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw....
.