1 Introduction
1.1 Definition and notation
Bernoulli polynomials \(B_n(x)\) form a sequence of polynomials indexed by a nonnegative integer \(n\). They are commonly introduced through an exponential generating function: for a complex variable \(t\) near \(0\), \[ \frac{t e^{xt}}{e^t-1}=\sum_{n=0}^{\infty} B_n(x)\frac{t^n}{n!}. \] The coefficients \(B_n(x)\) are polynomials in \(x\) of degree \(n\). In particular, evaluating at \(x=0\) yields the Bernoulli numbers \(B_n:=B_n(0)\).
1.2 Historical context and basic references
Bernoulli polynomials extend the classical work of Jacob Bernoulli on sums of powers. In later developments, they became standard tools in calculus of finite differences, asymptotic analysis, and the study of special functions. References typically include classical treatises on generating functions and analytic number theory texts that discuss the Hurwitz zeta function and the Euler–Maclaurin summation formula.
1.3 Relation to Bernoulli numbers
The Bernoulli numbers are a special case: \[ B_n = B_n(0). \] More generally, shifting the input \(x\) expresses \(B_n(x)\) in terms of Bernoulli numbers and binomial coefficients. This interplay is central in applications, since many series and asymptotic expansions involve \(B_n\) but are naturally organized by the polynomial refinement \(B_n(x)\).
2 Generating Functions
2.1 Exponential generating function
The defining exponential generating function \[ \frac{t e^{xt}}{e^t-1} \] encodes the entire family \(B_n(x)\). The factor \(e^{xt}\) accounts for the polynomial dependence on \(x\), while the remaining quotient \(\frac{t}{e^t-1}\) controls the normalization at \(x=0\) and the coefficients that become Bernoulli numbers.
2.2 Coefficient extraction and first polynomials
Expanding near \(t=0\) gives the first few polynomials: \[ B_0(x)=1,\quad B_1(x)=x-\tfrac12,\quad B_2(x)=x^2-x+\tfrac16, \] \[ B_3(x)=x^3-\tfrac32 x^2+\tfrac12 x,\quad B_4(x)=x^4-2x^3+x^2-\tfrac{1}{30}. \] These match the coefficients obtained by comparing powers of \(t\) in the generating series.
2.3 Alternative generating forms
Several equivalent forms are useful in computation. For example, rewriting with the substitution \(e^t-1=(e^{t/2})^2-(e^{t/2})^{-2}\) leads to forms emphasizing symmetry, while separating \(e^{xt}\) allows immediate identification of the shift behavior. In practice, any expression algebraically equivalent to the defining generating function yields the same coefficients \(B_n(x)\).
3 Core Properties
3.1 Derivative and Appell structure
Bernoulli polynomials constitute an Appell sequence. Differentiation lowers the index: \[ \frac{d}{dx}B_n(x)=n\,B_{n-1}(x),\qquad n\ge 1. \] This property is consistent with the presence of \(e^{xt}\) in the generating function: differentiating with respect to \(x\) multiplies the series by \(t\), shifting the index in the coefficient comparison.
3.2 Shift and difference relations
A fundamental relation comes from comparing the generating function at \(x+1\) with that at \(x\). One obtains the identity \[ B_n(x+1)-B_n(x)=n x^{n-1}, \] valid for \(n\ge 1\). This expresses \(B_n\) as a polynomial whose discrete difference recovers a monomial, linking Bernoulli polynomials directly to finite-difference calculus and power-sum formulas.
3.3 Reflection and symmetry identities
Bernoulli polynomials satisfy reflection-type formulas. A widely used identity is \[ B_n(1-x)=(-1)^n B_n(x), \] which follows from symmetry transformations of the generating function. Consequences include simplified evaluations at symmetric points and predictable sign behavior for odd or even degrees.
3.4 Degree, leading coefficient, and normalization
From the generating function, \(B_n(x)\) has degree \(n\) with leading coefficient \(1\). The normalization is fixed by \(B_n(0)=B_n\) and by the coefficient of \(t^n\) in \(\frac{t}{e^t-1}\). In particular, \(B_0(x)=1\) and \(B_1(x)=x-\tfrac12\) establish the standard convention for the first nontrivial polynomial.
4 Explicit Formulas and Computation
4.1 Low-degree expansions
For practical work, explicit expressions for small \(n\) are frequently listed. Using the generating function, one obtains successive polynomials such as \[ B_1(x)=x-\tfrac12,\quad B_2(x)=x^2-x+\tfrac16,\quad B_3(x)=x^3-\tfrac32 x^2+\tfrac12 x, \] and higher degrees can be generated similarly. These formulas are useful for checking identities and for illustrating general structural claims like degree and symmetry.
4.2 Expression via Bernoulli numbers
A standard representation connects the polynomial values to Bernoulli numbers: \[ B_n(x)=\sum_{k=0}^{n} \binom{n}{k} B_k\, x^{n-k}. \] This follows from expanding \(e^{xt}\) and using the generating function for \(B_k\). It provides a direct computational route: once the Bernoulli numbers are known, \(B_n(x)\) becomes an explicit polynomial in \(x\).
4.3 Recursive computation schemes
In addition to closed expansions, Bernoulli polynomials can be computed via recursion from their difference equation and derivative rule. For instance, using \[ B_n(x+1)=B_n(x)+n x^{n-1}, \] together with evaluations at a convenient starting point, one can build values and coefficients for successive \(n\). In algorithmic settings, such recursions often complement direct coefficient extraction from the generating function.
4.4 Summation/power-sum applications
The defining difference relation underlies power-sum formulas. Summing the identity \(B_n(x+1)-B_n(x)=n x^{n-1}\) over an arithmetic progression leads to expressions for sums of integer powers in terms of Bernoulli polynomials and numbers. Conceptually, the polynomial \(B_n(x)\) acts as a discrete antiderivative for \(x^{n-1}\), enabling closed forms for \(\sum k^{m}\).
5 Connections to Special Functions
5.1 Hurwitz zeta function relation
A central analytic connection is that Bernoulli polynomials encode values of the Hurwitz zeta function at negative integers. For \(n\ge 1\) and \(0<\Re(a)\), \[ \zeta(-n,a)=-\frac{B_{n+1}(a)}{n+1}. \] Thus, knowledge of \(B_{n}(a)\) provides zeta-function special values, and conversely the analytic continuation properties of the zeta function yield identities for Bernoulli polynomials.
5.2 Riemann zeta specializations
The Riemann zeta function appears as a special case of the Hurwitz zeta function. Taking \(a=1\) gives \[ \zeta(-n)= -\frac{B_{n+1}}{n+1}. \] Consequently, many classical results about \(\zeta(s)\) at nonpositive integers translate into statements about Bernoulli numbers and, via the polynomial relation, into corresponding formulas for \(B_n(x)\).
5.3 Analytic continuation aspects
While Bernoulli polynomials themselves are polynomials, the link to zeta functions brings analytic continuation into the picture. The regular values at negative integers emerge as finite expressions involving \(B_n(x)\). In effect, the generating-function machinery that defines \(B_n(x)\) also provides a bridge from elementary algebraic objects to analytic structures defined initially for \(\Re(s)>1\) and then extended elsewhere.
5.4 Fourier-type expansions and periodicity
Bernoulli polynomials connect to Fourier series of periodic functions in several ways. For suitable periodic extensions, one can express certain sawtooth- or spline-like functions using \(B_n(\{x\})\), where \(\{x\}\) denotes the fractional part. These expansions highlight how the polynomial’s symmetry and endpoint behavior govern the decay and harmonic content of the associated Fourier coefficients.
6 Integral and Series Representations
6.1 Integral representations
Bernoulli polynomials admit integral representations derived from their generating function. One common approach uses identities that rewrite \(\frac{t}{e^t-1}\) as an integral transform, then interchanges summation and integration to obtain a formula for \(B_n(x)\) in terms of weighted moments. Such representations are valuable for asymptotics and for establishing bounds.
6.2 Contour/generating-function derivations (survey)
In more advanced settings, contour methods express \(B_n(x)\) via coefficient extraction: \[ B_n(x)=\frac{n!}{2\pi i}\oint \frac{t e^{xt}}{e^t-1}\frac{dt}{t^{n+1}}, \] where the contour encircles the origin. Deforming the contour and accounting for poles at \(2\pi i k\) yields series representations linked to periodic sums and exponential terms. While different expositions choose different technical setups, the guiding idea is uniform: coefficients in a meromorphic generating function can be recovered by residues.
6.3 Series expansions involving \(B_n(x)\)
Besides residue-based series, Bernoulli polynomials appear in expansions of rational functions and in asymptotic developments of special functions. A typical pattern is that \(B_n(x)\) coefficients arise when one expands expressions in powers of a small parameter, particularly in contexts where Euler–Maclaurin-type corrections are needed.
6.4 Mean values and orthogonality-like identities
Although Bernoulli polynomials are not orthogonal in the usual \(L^2\) sense with a classical weight, they satisfy integral relations that resemble orthogonality for certain combinations. For example, integrating products against low-degree polynomials or using the Appell property leads to identities controlling moments of \(B_n(x)\). These “orthogonality-like” statements are often tailored to the specific application, such as periodic summation or error estimates.
7 Applications in Mathematical Analysis
7.1 Euler–Maclaurin summation (overview)
The Euler–Maclaurin summation formula uses Bernoulli polynomials to correct discrete sums by continuous integrals. In simplified terms, the correction terms involve Bernoulli numbers (or Bernoulli polynomials evaluated at endpoints), producing an asymptotic expansion for \[ \sum_{k=a}^{b} f(k) \] in terms of derivatives of \(f\). This turns the polynomial’s difference structure into a systematic method for tracking discretization error.
7.2 Asymptotic expansions in smooth sums
When \(f\) is sufficiently smooth, repeated integration by parts in the Euler–Maclaurin framework generates an asymptotic series where the coefficients depend on \(B_n(x)\). As \(n\) grows, these coefficients reflect higher-order derivatives of \(f\), while growth and remainder behavior are governed by analytic properties of \(f\) and by the size of Bernoulli numbers/polynomials.
7.3 Summation of periodic functions
Bernoulli polynomials can be used to evaluate sums of periodic or nearly periodic functions by expressing the periodic behavior through fractional-part functions. The reflection identity and the behavior at integer shifts help determine how endpoints contribute. Such techniques often yield closed forms or rapidly convergent expansions for classes of trigonometric or piecewise smooth periodic functions.
7.4 Approximations and error terms (conceptual)
In practical computations, truncated Euler–Maclaurin expansions provide approximations whose accuracy depends on the smoothness of the function and the magnitude of the omitted Bernoulli-related terms. While the exact remainder depends on the analytic context, Bernoulli polynomials function as the universal “correction kernel” that dictates how rapidly the discrete sum approaches the integral-based approximation.
8 Functional Identities and Transformations
8.1 Polynomial identities from generating functions
Many identities follow directly by manipulating the generating function. For example, substitutions like \(x\mapsto x+y\) correspond to multiplying generating functions by \(e^{yt}\), producing binomial-type expansions. Such methods yield systematic polynomial relations without requiring case-by-case algebra.
8.2 Multiplication/scaling formulas
Bernoulli polynomials satisfy scaling identities that relate \(B_n(cx)\) to linear combinations of Bernoulli polynomials at fractional shifts. These formulas arise by comparing generating functions after replacing \(t\) with \(ct\) and adjusting the denominator \(e^{ct}-1\). Multiplication theorems are especially useful when evaluating sums over arithmetic progressions.
8.3 Composition/shift operators
The Appell and difference properties can be packaged into operator statements: shifting \(x\) corresponds to an exponential operator acting on polynomials, while differentiation corresponds to lowering the index. This operator viewpoint clarifies why Bernoulli polynomials behave well under discrete integration and why they serve naturally in finite-difference and summation methods.
8.4 Boundary value evaluations (e.g., \(x=0,1\))
Evaluations at special arguments, particularly \(x=0\) and \(x=1\), simplify many expressions. For instance, using the symmetry relation one obtains \[ B_n(1)=(-1)^n B_n(0)=(-1)^n B_n. \] Endpoint evaluations often appear in formulas derived from Euler–Maclaurin summation, where corrections depend on values at the boundaries of the summation interval.
9 Special Cases and Examples
9.1 Bernoulli polynomials at integers
At integer inputs, \(B_n(m)\) can be expressed using Bernoulli numbers and polynomial shifts. The difference equation implies that values along consecutive integers are linked through monomials, enabling explicit computation. In many applications, integer evaluations determine zeta special values and the constant terms in asymptotic expansions.
9.2 Values at rational arguments (overview)
For rational \(x=p/q\), Bernoulli polynomials admit representations involving finite sums and special constants through connections to Hurwitz zeta values. Although explicit expressions may require additional work, the key takeaway is that rational inputs correspond to zeta-function values at negative integers with parameter \(a=p/q\).
9.3 Symmetric points and simplified forms
At points related by reflection, such as \(x\) and \(1-x\), the symmetry identity forces predictable transformations: \[ B_n(1-x)=(-1)^n B_n(x). \] This often yields simplifications for odd \(n\) (antisymmetry around \(x=\tfrac12\)) and for even \(n\) (symmetry). Such behavior also makes \(x=\tfrac12\) a natural evaluation point for many derived expressions.
9.4 Worked examples for small \(n\)
Using the low-degree formulas, one can quickly illustrate typical evaluations. For example:
- At \(x=0\), \(B_1(0)=-\tfrac12\), \(B_2(0)=\tfrac16\), and \(B_3(0)=0\).
- At \(x=1\), \(B_1(1)=\tfrac12\), \(B_2(1)=\tfrac16\), and \(B_3(1)=0\), consistent with the reflection relation.
- At \(x=\tfrac12\), \(B_1(\tfrac12)=0\) and \(B_2(\tfrac12)=-\tfrac{1}{12}\), showing the role of symmetry and the normalization chosen in the generating function.