Bernoulli polynomials
Bernoulli polynomials are a sequence of polynomials B_n(x) of degree n, defined by the generating function \frac{t e^{xt}}{e^t - 1} = \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}, which generalize the Bernoulli numbers B_n = B_n(0) and play a fundamental role in number theory and analysis for expressing sums of powers of integers.[1][2] The Bernoulli numbers were introduced by Jacob Bernoulli in 1713 in his work Ars Conjectandi to solve problems involving sums of consecutive powers, while the polynomials were introduced by Leonhard Euler in 1738 through the explicit generating function.[2][1] Key properties include the explicit formula B_n(x) = \sum_{k=0}^n \binom{n}{k} B_{n-k} x^k, which expresses them in terms of Bernoulli numbers, and the recurrence relation B_n(x+1) - B_n(x) = n x^{n-1}, useful for deriving the formula for the sum of the first m n-th powers as \sum_{k=1}^m k^n = \frac{1}{n+1} \left[ B_{n+1}(m+1) - B_{n+1} \right].[3][1][2] Additional notable features are their symmetry B_n(1-x) = (-1)^n B_n(x) for all non-negative integers n, the differentiation rule \frac{d}{dx} B_n(x) = n B_{n-1}(x), and their Fourier series expansion B_n(x) = -\frac{n!}{(2\pi i)^n} \sum_{k \neq 0} \frac{e^{2\pi i k x}}{k^n} for non-integer x.[3][1] These polynomials appear in diverse applications, including the Euler-Maclaurin formula for approximating sums by integrals, the study of zeta functions, and asymptotic expansions in special functions.[1][3]History
Origins and Bernoulli Numbers
The Bernoulli numbers were first introduced by the Swiss mathematician Jacob Bernoulli in his posthumously published treatise Ars Conjectandi in 1713, where he developed them as a tool for deriving explicit formulas for the sums of powers of the first n positive integers, such as \sum_{k=1}^n k^p. Bernoulli also introduced the Bernoulli polynomials B_n(x) in this work to express these sums as \sum_{k=1}^m k^n = \frac{1}{n+1} [B_{n+1}(m+1) - B_{n+1}(0)].[1] Bernoulli's approach involved identifying a recursive pattern in these sums, leading to a general expression that incorporated a sequence of rational coefficients now known as the Bernoulli numbers.[4] This work built on earlier efforts by mathematicians like Johann Faulhaber, but Bernoulli's systematic treatment marked a significant advance in understanding power sums through combinatorial methods.[5] Despite their foundational role, the numbers are named after Jacob Bernoulli, even though subsequent key developments, including their broader applications and generalizations, were advanced by later figures such as Leonhard Euler.[3] In the 18th century, Euler incorporated Bernoulli numbers into the Euler-Maclaurin summation formula, independently discovered around 1735 alongside Colin Maclaurin, to approximate sums by integrals with correction terms derived from these numbers.[6] This formula highlighted the numbers' utility in bridging discrete sums and continuous integrals, drawing an analogy between finite differences in discrete calculus and derivatives in continuous analysis, and built upon the earlier Bernoulli polynomials for power sums.[7] The Bernoulli polynomials themselves appear as special cases evaluated at specific points, such as x = 0 or 1, preserving their recursive and combinatorial properties in the context of the Euler-Maclaurin formula.[8]Development and Applications
Leonhard Euler further developed the Bernoulli polynomials, providing their generating function in 1738 and exploring them for arbitrary x in his Institutiones calculi differentialis (1755), to facilitate the summation of power series and applications in the Euler-Maclaurin formula, which approximates definite integrals by finite sums and provides asymptotic expansions for divergent series.[1] [3] This extension allowed for more flexible expressions in analyzing sums of powers, such as \sum_{k=1}^n k^m, as polynomials in n.[9] In the 19th century, mathematicians advanced the study of Bernoulli polynomials, with J.L. Raabe coining the term "Bernoulli polynomials" in 1851. They were used in the expansion of trigonometric series and the study of periodic functions, particularly in connection with zeta function evaluations.[8] [9] The 20th century saw significant developments through Gian-Carlo Rota's formalization of umbral calculus in the 1970s, which provided a unified operator-based framework for manipulating Bernoulli polynomials alongside other special functions like Hermite and Laguerre polynomials.[10] Rota's approach, using linear functionals on polynomial rings, revealed deep structural similarities and facilitated derivations of identities for these sequences without explicit computations.[11] Early applications of Bernoulli polynomials extended to asymptotic analysis, where they underpin expansions in the Euler-Maclaurin formula for approximating integrals and sums in physics and engineering contexts.[12] In number theory, extensions of the von Staudt–Clausen theorem to Bernoulli polynomials, such as those for Hurwitz zeta values, determine denominators and modular properties, influencing results on L-functions and arithmetic progressions.[13]Definitions
Generating Function
The exponential generating function for the Bernoulli polynomials B_n(x) is \frac{t e^{x t}}{e^t - 1} = \sum_{n=0}^{\infty} B_n(x) \frac{t^n}{n!}, valid for |t| < 2\pi.[14] This series defines the polynomials B_n(x) as the coefficients of t^n / n! in the Laurent series expansion of the left-hand side around t = 0.[14] This generating function arises from the corresponding exponential generating function for the Bernoulli numbers, \frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}, also valid for |t| < 2\pi, where B_n = B_n(0).[14] Multiplying by the exponential shift factor e^{x t} incorporates the parameter x, yielding the more general form and allowing the extraction of B_n(x) for arbitrary x.[14] This construction was first introduced by Euler in 1738.[1] The structure of the generating function, expressed as e^{x t} g(t) with g(t) = t / (e^t - 1), identifies the Bernoulli polynomials as an Appell sequence.[1] Consequently, they satisfy the translation formula B_n(x + y) = \sum_{k=0}^n \binom{n}{k} B_k(x) y^{n-k}, which derives from equating the generating functions for x + y and for x, then multiplying by e^{y t} and comparing coefficients via the binomial theorem.[15]Explicit Formula
The standard closed-form expression for the Bernoulli polynomials B_n(x) is given by B_n(x) = \sum_{k=0}^n \binom{n}{k} B_k x^{n-k}, where B_k denotes the kth Bernoulli number. This formula arises directly from the generating function for the Bernoulli polynomials, \frac{t e^{xt}}{e^t - 1} = \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}. Rewriting the left side as e^{xt} \cdot \frac{t}{e^t - 1} and expanding each factor as a power series—e^{xt} = \sum_{m=0}^\infty x^m \frac{t^m}{m!} and \frac{t}{e^t - 1} = \sum_{k=0}^\infty B_k \frac{t^k}{k!}—the coefficient of \frac{t^n}{n!} in the product is obtained via the Cauchy product of the series, which simplifies to the binomial sum above using the binomial theorem. The expression assumes the conventional choice B_1 = -\frac{1}{2} for the Bernoulli numbers, which ensures B_1(x) = x - \frac{1}{2} and aligns with the generating function definition. In the alternative convention B_1^+ = +\frac{1}{2}, the corresponding polynomials B_n^+(x) satisfy B_n^+(x) = (-1)^{n+1} B_n(1 - x) for n \geq 2, altering the explicit sum to use B_k^+ instead; this impacts applications like the Euler-Maclaurin formula by changing the sign for odd-degree terms. An alternative closed-form representation expresses the Bernoulli polynomials in terms of the Hurwitz zeta function: B_n(x) = -n \zeta(1 - n, x), \quad n \geq 1, \ \operatorname{Re} x > 0, where \zeta(s, x) = \sum_{k=0}^\infty (k + x)^{-s} for \operatorname{Re} s > 1, extended analytically. This form highlights connections to zeta function regularization and asymptotic expansions.Representations
Operator Forms
One representation of the Bernoulli polynomials arises from the generating function through a differential operator. Specifically, the nth Bernoulli polynomial is given by the nth derivative of the generating function evaluated at zero: B_n(x) = \left. \frac{d^n}{dt^n} \left( \frac{t e^{x t}}{e^t - 1} \right) \right|_{t=0}, where the notation \left( f(t) \right)^{\underline{n}} |_{t=0} denotes this operation. This form facilitates algebraic manipulations in operational calculus and connects directly to the exponential generating function \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!} = \frac{t e^{x t}}{e^t - 1}.[1] A key connection to finite difference operators is provided by the forward difference \Delta f(x) = f(x+1) - f(x). For Bernoulli polynomials, this yields \Delta B_n(x) = n x^{n-1}, which highlights their role in discretizing derivatives and appears in summation formulas like Euler-Maclaurin. This relation holds for n \geq 1 and underscores the polynomials' utility in approximating integrals by sums.[1] In umbral calculus, Bernoulli polynomials admit an elegant operator interpretation where B_n(x) = (x + B)^n, with B denoting the umbral variable satisfying B^k = B_k for the kth Bernoulli number B_k = B_k(0). This formal expansion treats the polynomials as if x and B commute in the umbral algebra, enabling mnemonic derivations of identities such as translation formulas B_n(x + y) = \sum_{k=0}^n \binom{n}{k} B_k(x) y^{n-k}. The approach leverages the structure of finite operator calculus to unify various polynomial sequences.[1]Integral Representations
One prominent integral representation of the Bernoulli polynomials arises from their generating function \frac{t e^{xt}}{e^t - 1} = \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}, valid for |t| < 2\pi. By extracting the coefficient of t^n/n! using Cauchy's residue theorem, with a counterclockwise contour C encircling the origin and avoiding the poles of e^t - 1 at t = 2\pi i k for integers k \neq 0, the representation is B_n(x) = \frac{n!}{2\pi i} \oint_C \frac{e^{x z}}{(e^z - 1) z^{n+1}} \, dz. This contour integral provides a global analytic expression useful for studying properties in the complex plane.[14] Another contour integral representation, known as the Mellin–Barnes form, expresses the Bernoulli polynomials in terms of the Gamma function through the identity \pi / \sin(\pi t) = \Gamma(t) \Gamma(1 - t): B_n(x) = \frac{1}{2\pi i} \int_{-c - i\infty}^{-c + i\infty} (x + t)^n \left( \frac{\pi}{\sin(\pi t)} \right)^2 \, dt, where the vertical contour satisfies $0 < c < 1. This form links Bernoulli polynomials to the Hurwitz zeta function via \zeta(1 - n, x) = -B_n(x)/n for positive integers n \geq 2, facilitating connections to analytic number theory.[16] Fourier-type integral representations offer real-line expressions that serve as precursors to the full Fourier series expansion of Bernoulli polynomials on [0, 1). For even degrees, B_{2n}(x) = (-1)^{n+1} 2n \int_0^\infty \frac{\cos(2\pi x) - e^{-2\pi t}}{\cosh(2\pi t) - \cos(2\pi x)} t^{2n-1} \, dt, valid for n = 1, 2, \dots and $0 < \Re x < 1. Similarly, for odd degrees greater than 1, B_{2n+1}(x) = (-1)^{n+1} (2n+1) \int_0^\infty \frac{\sin(2\pi x t)}{\cosh(2\pi t) - \cos(2\pi x)} t^{2n} \, dt, under the same conditions. These integrals highlight the periodic nature of Bernoulli polynomials and aid in deriving their Fourier series.[16] These integral representations are particularly valuable for asymptotic analysis, where saddle-point methods applied to the contour integrals yield expansions for large |x| or high degrees. For instance, deforming the contour in the generating function integral to pass through a saddle point provides uniform asymptotic approximations, such as B_\nu(z) \sim z^\nu \sum_{k=0}^\infty c_k(\nu) z^{-k} for large |z| with \arg z bounded away from the negative real axis, enabling precise estimates in applications like summation formulas.[17]Properties
Recurrence Relations
The Bernoulli polynomials satisfy several recurrence relations that facilitate their computation and analysis. A fundamental relation is the difference equation B_n(x+1) - B_n(x) = n x^{n-1}, valid for all nonnegative integers n and real x. This identity can be derived from the generating function \frac{t e^{xt}}{e^t - 1} = \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}. Substituting x+1 yields \frac{t e^{(x+1)t}}{e^t - 1} = e^t \cdot \frac{t e^{xt}}{e^t - 1}, so the difference of generating functions is (e^t - 1) \cdot \frac{t e^{xt}}{e^t - 1} = t e^{xt}. Expanding the right side as t \sum_{k=0}^\infty \frac{(xt)^k}{k!} = \sum_{n=1}^\infty n x^{n-1} \frac{t^n}{n!} and equating coefficients gives the recurrence. An integral recurrence follows from the derivative property B_n'(x) = n B_{n-1}(x), which is obtained by differentiating the generating function with respect to x. Integrating both sides yields \int B_{n-1}(x) \, dx = \frac{1}{n} B_n(x) + C. For the definite integral over an interval of length 1, \int_0^1 B_n(x + t) \, dt = \frac{B_{n+1}(x+1) - B_{n+1}(x)}{n+1}. This holds because the substitution u = x + t transforms the left side to \int_x^{x+1} B_n(u) \, du, and applying the antiderivative gives the right side. These relations enable inductive computation of the polynomials. Starting from B_0(x) = 1, the explicit summation formula B_n(x) = \sum_{k=0}^n \binom{n}{k} B_k x^{n-k}, where B_k = B_k(0) are the Bernoulli numbers, allows determination of higher-degree terms using previously computed lower-degree polynomials. This sum is derived by expanding the generating function \frac{t e^{xt}}{e^t - 1} = \frac{t}{e^t - 1} \cdot e^{xt} = \left( \sum_{k=0}^\infty B_k \frac{t^k}{k!} \right) \left( \sum_{m=0}^\infty \frac{(xt)^m}{m!} \right) and collecting coefficients. The difference recurrence then verifies or extends these computations for shifted arguments.Symmetries and Translations
The Bernoulli polynomials exhibit a fundamental translation property that reflects their structure as a binomial transform of the Bernoulli numbers. Specifically, for nonnegative integers n and real numbers x, y, B_n(x + y) = \sum_{k=0}^n \binom{n}{k} B_k(x) y^{n-k}. This relation arises directly from the generating function \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!} = \frac{t e^{x t}}{e^t - 1}; substituting x \to x + y yields \frac{t e^{(x+y) t}}{e^t - 1} = e^{y t} \cdot \frac{t e^{x t}}{e^t - 1}, and expanding e^{y t} = \sum_{m=0}^\infty \frac{(y t)^m}{m!} produces the binomial expansion in the coefficients of \frac{t^n}{n!}.[18] As a member of the Appell sequence of polynomials, this translation invariance underscores the Bernoulli polynomials' role in generalizing sequences satisfying similar shift properties.[19] A key symmetry of the Bernoulli polynomials is captured by the reflection formula B_n(1 - x) = (-1)^n B_n(x), which holds for all nonnegative integers n. This identity implies that the polynomials are even or odd functions with respect to the point x = 1/2, depending on the parity of n: for even n, B_n(1/2 + z) = B_n(1/2 - z), while for odd n, B_n(1/2 + z) = -B_n(1/2 - z). The formula applies uniformly, including for n=1 where B_1(x) = x - 1/2 satisfies B_1(1 - x) = -B_1(x), though n=1 is exceptional in that B_1(1) = 1/2 \neq -1/2 = B_1(0), unlike higher degrees where B_n(1) = B_n(0) for n \neq 1. For odd degrees n > 1, the reflection formula combines with the vanishing of the corresponding Bernoulli numbers B_n(0) = 0 to ensure antisymmetry around x=1/2 without endpoint equality at integers beyond the n=1 case.[19] The reflection formula can be derived from the generating function by substituting x \to 1 - x, yielding \frac{t e^{(1-x) t}}{e^t - 1} = e^{t} \cdot \frac{t e^{-x t}}{e^t - 1}. Multiplying numerator and denominator in the fraction by e^{-t} gives \frac{t e^{-x t}}{1 - e^{-t}}, and recognizing that \frac{t}{1 - e^{-t}} = -\frac{t e^{t}}{e^{t} - 1} \cdot (-1) with t \to -t relates the series to \sum_{n=0}^\infty (-1)^n B_n(x) \frac{t^n}{n!}, confirming the sign alternation. This generating function approach highlights the intrinsic symmetry embedded in the exponential structure defining the polynomials.Differences and Derivatives
The derivative of the Bernoulli polynomial B_n(x) satisfies the relation B_n'(x) = n B_{n-1}(x) for n \geq 1. This formula arises from term-by-term differentiation of the generating function \frac{t e^{xt}}{e^t - 1} = \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}, yielding a factor of t that shifts the degree by one, and holds in the complex domain for \Re(v) > -1 and |\arg z| < \pi when extended to non-integer orders.[17] The forward difference \Delta B_n(x) = B_n(x+1) - B_n(x) simplifies to \Delta B_n(x) = n x^{n-1} for n \geq 1. This property, which encodes the defining role of Bernoulli polynomials in discrete calculus, also extends to complex arguments under the same conditions as the derivative formula. It directly implies that \frac{B_n(x)}{n} serves as an antiderivative (indefinite sum) for the monomial x^{n-1} under the forward difference operator.[17] Higher-order forward differences \Delta^k B_n(x) for $1 \leq k \leq n can be computed iteratively from the first difference, resulting in polynomials of degree n-k whose leading term is the falling factorial coefficient n^{\underline{k}} x^{n-k} = n(n-1)\cdots(n-k+1) x^{n-k}, accompanied by lower-degree terms arising from the binomial expansions in repeated applications of \Delta. The explicit form for the translation underlying these differences is B_n(x + m) = B_n(x) + n \sum_{j=0}^{m-1} (x + j)^{n-1}, allowing computation of \Delta^k B_n(x) via inclusion-exclusion on multiple translations; for example, \Delta^2 B_n(x) = n[(x+1)^{n-1} - x^{n-1}] = n(n-1)x^{n-2} + \frac{n(n-1)(n-2)}{2} x^{n-3} + \cdots. These higher differences connect Bernoulli polynomials to the falling factorial basis (x)_l = x(x-1)\cdots(x-l+1), in which the forward difference acts exactly as \Delta (x)_l = l (x)_{l-1}, mirroring differentiation on powers. Specifically, the expansion B_n(x) = B_n + \sum_{k=1}^n \binom{n}{k} S(n-1, k-1) x^k, combined with the change-of-basis formula x^k = \sum_{l=0}^k S(k,l) (x)_l (where S(k,l) are Stirling numbers of the second kind), expresses B_n(x) as a linear combination of falling factorials with coefficients involving Stirling numbers, facilitating computations in discrete settings where the falling factorial basis diagonalizes the difference operator.[17][20] This structure under differences interprets Bernoulli polynomials as tools for discrete integration by parts (summation by parts). In the summation formula \sum_{j=a}^b u_j \Delta v_j = u_{b+1} v_{b+1} - u_a v_a - \sum_{j=a}^b \Delta u_j \, v_{j+1}, setting v_j = \frac{B_n(j)}{n} yields \Delta v_j = j^{n-1}, allowing sums of powers \sum j^{n-1} to be expressed using boundary terms involving Bernoulli polynomials, analogous to how x^{n-1}/(n-1) integrates (n-1) x^{n-2} in the continuous case via integration by parts. This enables closed-form evaluation of power sums and underpins applications in numerical analysis and combinatorial identities.Explicit Forms
Low-Degree Expressions
The Bernoulli polynomials for low degrees provide concrete examples that illustrate their structure and utility in computations. These polynomials are monic, meaning the coefficient of the leading term x^n is always 1 for degree n, and their constant terms correspond to the Bernoulli numbers B_n. The following explicit expressions for degrees 0 through 6 are derived from the standard definition and can be verified by expanding the generating function \frac{te^{xt}}{e^t - 1} = \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!} up to the respective order or by applying the explicit formula B_n(x) = \sum_{k=0}^n \binom{n}{k} B_k x^{n-k}, where B_k are the Bernoulli numbers.[1][1] For clarity, the polynomials are presented in the table below:| Degree n | B_n(x) |
|---|---|
| 0 | $1 |
| 1 | x - \frac{1}{2} |
| 2 | x^2 - x + \frac{1}{6} |
| 3 | x^3 - \frac{3}{2} x^2 + \frac{1}{2} x |
| 4 | x^4 - 2 x^3 + x^2 - \frac{1}{30} |
| 5 | x^5 - \frac{5}{2} x^4 + \frac{5}{3} x^3 - \frac{1}{6} x |
| 6 | x^6 - 3 x^5 + \frac{5}{2} x^4 - \frac{1}{2} x^2 + \frac{1}{42} |