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| 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 [[trigonometric function|sine and cosine functions]].
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| [[Image:Bernoulli polynomials.svg|thumb|right|Bernoulli polynomials]]
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| ==Representations==
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| The Bernoulli polynomials ''B''<sub>''n''</sub> admit a variety of different [[representation (mathematics)|representations]]. Which among them should be taken to be the definition may depend on one's purposes.
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| ===Explicit formula===
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| :<math>B_n(x) = \sum_{k=0}^n {n \choose k} b_{n-k} x^k,</math>
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| for ''n'' ≥ 0, where ''b''<sub>''k''</sub> are the [[Bernoulli number]]s.
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| ===Generating functions===
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| The [[generating function]] for the Bernoulli polynomials is
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| :<math>\frac{t e^{xt}}{e^t-1}= \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}.</math>
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| The generating function for the Euler polynomials is
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| :<math>\frac{2 e^{xt}}{e^t+1}= \sum_{n=0}^\infty E_n(x) \frac{t^n}{n!}.</math>
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| ===Representation by a differential operator===
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| The Bernoulli polynomials are also given by
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| :<math>B_n(x)={D \over e^D -1} x^n</math>
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| where ''D'' = ''d''/''dx'' is differentiation with respect to ''x'' and the fraction is expanded as a [[formal power series]]. It follows that
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| :<math>\int _a^x B_n (u) ~du = \frac{B_{n+1}(x) - B_{n+1}(a)}{n+1} ~.</math>
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| cf. [[#Integrals]] below.
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| ===Representation by an integral operator===
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| The Bernoulli polynomials are the unique polynomials determined by
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| :<math>\int_x^{x+1} B_n(u)\,du = x^n.</math>
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| The [[integral transform]]
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| :<math>(Tf)(x) = \int_x^{x+1} f(u)\,du</math>
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| on polynomials ''f'', simply amounts to
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| :<math>
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| \begin{align}
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| (Tf)(x) = {e^D - 1 \over D}f(x) & {} = \sum_{n=0}^\infty {D^n \over (n+1)!}f(x) \\
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| & {} = f(x) + {f'(x) \over 2} + {f''(x) \over 6} + {f'''(x) \over 24} + \cdots ~.
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| \end{align}
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| </math>
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| This can be used to produce the [[#Inversion]] formulas below.
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| ==Another explicit formula==
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| An explicit formula for the Bernoulli polynomials is given by
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| :<math>B_m(x)=
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| \sum_{n=0}^m \frac{1}{n+1}
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| \sum_{k=0}^n (-1)^k {n \choose k} (x+k)^m.</math>
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| Note the remarkable similarity to the globally convergent series expression for the [[Hurwitz zeta function]]. Indeed, one has
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| :<math>B_n(x) = -n \zeta(1-n,x)</math>
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| 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''.
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| The inner sum may be understood to be the ''n''th [[forward difference]] of ''x''<sup>''m''</sup>; that is,
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| :<math>\Delta^n x^m = \sum_{k=0}^n (-1)^{n-k} {n \choose k} (x+k)^m</math>
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| where Δ is the [[forward difference operator]]. Thus, one may write
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| :<math>B_m(x)= \sum_{n=0}^m \frac{(-1)^n}{n+1} \Delta^n x^m. </math>
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| This formula may be derived from an identity appearing above as follows. Since the forward difference operator Δ equals
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| :<math>\Delta = e^D - 1\,</math>
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| where ''D'' is differentiation with respect to ''x'', we have, from the [[Mercator series]]
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| :<math>{D \over e^D - 1} = {\log(\Delta + 1) \over \Delta} = \sum_{n=0}^\infty {(-\Delta)^n \over n+1}.</math>
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| As long as this operates on an ''m''th-degree polynomial such as ''x''<sup>''m''</sup>, one may let ''n'' go from 0 only up to ''m''.
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| An integral representation for the Bernoulli polynomials is given by the [[Nörlund–Rice integral]], which follows from the expression as a finite difference.
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| An explicit formula for the Euler polynomials is given by
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| :<math>E_m(x)=
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| \sum_{n=0}^m \frac{1}{2^n}
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| \sum_{k=0}^n (-1)^k {n \choose k} (x+k)^m\,.</math>
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| This may also be written in terms of the [[Euler number]]s ''E''<sub>''k''</sub> as
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| :<math>E_m(x)=
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| \sum_{k=0}^m {m \choose k} \frac{E_k}{2^k}
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| \left(x-\frac{1}{2}\right)^{m-k} \,.</math>
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| ==Sums of ''p''th powers==
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| We have
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| :<math>\sum_{k=0}^{x} k^p = \frac{B_{p+1}(x+1)-B_{p+1}(0)}{p+1}.</math>
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| See [[Faulhaber's formula]] for more on this.
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| ==The Bernoulli and Euler numbers==
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| The [[Bernoulli number]]s are given by <math>B_n=B_n(0).</math>
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| An alternate convention defines the Bernoulli numbers as <math>B_n=B_n(1)</math>. This definition gives B<sub>''n''</sub> = −''n''ζ(1 − ''n'') where for ''n'' = 0 and ''n'' = 1 the expression −''n''ζ(1 − ''n'') is to be understood as
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| lim<sub>''x'' → ''n''</sub> −''x''ζ(1 − ''x'').
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| The two conventions differ only for ''n'' = 1 since B<sub>1</sub>(1) = 1/2 = −B<sub>1</sub>(0).
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| The [[Euler number]]s are given by <math>E_n=2^nE_n(1/2).</math>
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| ==Explicit expressions for low degrees==
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| The first few Bernoulli polynomials are:
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| :<math>B_0(x)=1\,</math>
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| :<math>B_1(x)=x-1/2\,</math>
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| :<math>B_2(x)=x^2-x+1/6\,</math>
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| :<math>B_3(x)=x^3-\frac{3}{2}x^2+\frac{1}{2}x\,</math>
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| :<math>B_4(x)=x^4-2x^3+x^2-\frac{1}{30}\,</math>
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| :<math>B_5(x)=x^5-\frac{5}{2}x^4+\frac{5}{3}x^3-\frac{1}{6}x\,</math>
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| :<math>B_6(x)=x^6-3x^5+\frac{5}{2}x^4-\frac{1}{2}x^2+\frac{1}{42}.\,</math>
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| The first few Euler polynomials are
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| :<math>E_0(x)=1\,</math>
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| :<math>E_1(x)=x-1/2\,</math>
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| :<math>E_2(x)=x^2-x\,</math>
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| :<math>E_3(x)=x^3-\frac{3}{2}x^2+\frac{1}{4}\,</math>
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| :<math>E_4(x)=x^4-2x^3+x\,</math>
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| :<math>E_5(x)=x^5-\frac{5}{2}x^4+\frac{5}{2}x^2-\frac{1}{2}\,</math>
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| :<math>E_6(x)=x^6-3x^5+5x^3-3x.\,</math>
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| ==Maximum and minimum==
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| At higher ''n'', the amount of variation in ''B''<sub>''n''</sub>(''x'') between ''x'' = 0 and ''x'' = 1 gets large. For instance,
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| :<math>B_{16}(x)=x^{16}-8x^{15}+20x^{14}-\frac{182}{3}x^{12}+\frac{572}{3}x^{10}-429x^8+\frac{1820}{3}x^6
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| -\frac{1382}{3}x^4+140x^2-\frac{3617}{510}</math>
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| 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]]<ref>D.H. Lehmer, "On the Maxima and Minima of Bernoulli Polynomials", ''[[American Mathematical Monthly]]'', volume 47, pages 533–538 (1940)</ref> showed that the maximum value of ''B''<sub>''n''</sub>(''x'') between 0 and 1 obeys | |
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| :<math>M_n < \frac{2n!}{(2\pi)^n}</math>
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| unless ''n'' is 2 modulo 4, in which case
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| :<math>M_n = \frac{2\zeta(n)n!}{(2\pi)^n}</math>
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| (where <math>\zeta(x)</math> is the [[Riemann zeta function]]), while the minimum obeys
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| :<math>m_n > \frac{-2n!}{(2\pi)^n}</math>
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| unless ''n'' is 0 modulo 4, in which case
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| :<math>m_n = \frac{-2\zeta(n)n!}{(2\pi)^n}.</math>
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| These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well.
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| ==Differences and derivatives==
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| The Bernoulli and Euler polynomials obey many relations from [[umbral calculus]]:
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| :<math>\Delta B_n(x) = B_n(x+1)-B_n(x)=nx^{n-1},\,</math>
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| :<math>\Delta E_n(x) = E_n(x+1)-E_n(x)=2(x^n-E_n(x)).\,</math>
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| (Δ is the [[forward difference operator]]).
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| These [[polynomial sequence]]s are [[Appell sequence]]s:
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| :<math>B_n'(x)=nB_{n-1}(x),\,</math>
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| :<math>E_n'(x)=nE_{n-1}(x).\,</math>
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| ===Translations===
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| :<math>B_n(x+y)=\sum_{k=0}^n {n \choose k} B_k(x) y^{n-k}</math>
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| :<math>E_n(x+y)=\sum_{k=0}^n {n \choose k} E_k(x) y^{n-k}</math>
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| These identities are also equivalent to saying that these polynomial sequences are [[Appell sequence]]s. ([[Hermite polynomials]] are another example.)
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| ===Symmetries===
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| :<math>B_n(1-x)=(-1)^nB_n(x),\quad n \ge 0,</math>
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| :<math>E_n(1-x)=(-1)^n E_n(x)\,</math>
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| :<math>(-1)^n B_n(-x) = B_n(x) + nx^{n-1}\,</math>
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| :<math>(-1)^n E_n(-x) = -E_n(x) + 2x^n\,</math>
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| [[Zhi-Wei Sun]] and Hao Pan <ref>{{cite journal |author1=Zhi-Wei Sun |author2=Hao Pan |journal=Acta Arithmetica |volume=125 |year=2006 |pages=21–39 |title=Identities concerning Bernoulli and Euler polynomials |arxiv=math/0409035}}</ref> established the following surprising symmetry relation: If ''r'' + ''s'' + ''t'' = ''n'' and ''x'' + ''y'' + ''z'' = 1, then
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| :<math>r[s,t;x,y]_n+s[t,r;y,z]_n+t[r,s;z,x]_n=0,</math>
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| where
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| :<math>[s,t;x,y]_n=\sum_{k=0}^n(-1)^k{s \choose k}{t\choose {n-k}}
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| B_{n-k}(x)B_k(y).</math>
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| ==Fourier series==
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| The [[Fourier series]] of the Bernoulli polynomials is also a [[Dirichlet series]], given by the expansion
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| :<math>B_n(x) = -\frac{n!}{(2\pi i)^n}\sum_{k\not=0 }\frac{e^{2\pi ikx}}{k^n}= -2 n! \sum_{k=1}^{\infty} \frac{\cos\left(2 k \pi x- \frac{n \pi} 2 \right)}{(2 k \pi)^n}.</math>
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| Note the simple large ''n'' limit to suitably scaled trigonometric functions.
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| This is a special case of the analogous form for the [[Hurwitz zeta function]]
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| :<math>B_n(x) = -\Gamma(n+1) \sum_{k=1}^\infty
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| \frac{ \exp (2\pi ikx) + e^{i\pi n} \exp (2\pi ik(1-x)) } { (2\pi ik)^n }. </math>
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| This expansion is valid only for 0 ≤ ''x'' ≤ 1 when ''n'' ≥ 2 and is valid for 0 < ''x'' < 1 when ''n'' = 1.
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| The Fourier series of the Euler polynomials may also be calculated. Defining the functions
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| :<math>C_\nu(x) = \sum_{k=0}^\infty
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| \frac {\cos((2k+1)\pi x)} {(2k+1)^\nu}</math>
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| and
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| :<math>S_\nu(x) = \sum_{k=0}^\infty
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| \frac {\sin((2k+1)\pi x)} {(2k+1)^\nu}</math>
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| for <math>\nu > 1</math>, the Euler polynomial has the Fourier series | |
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| :<math>C_{2n}(x) = \frac{(-1)^n}{4(2n-1)!}
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| \pi^{2n} E_{2n-1} (x)</math>
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| and
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| :<math>S_{2n+1}(x) = \frac{(-1)^n}{4(2n)!}
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| \pi^{2n+1} E_{2n} (x).</math>
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| Note that the <math>C_\nu</math> and <math>S_\nu</math> are odd and even, respectively:
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| :<math>C_\nu(x) = -C_\nu(1-x)</math>
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| and
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| :<math>S_\nu(x) = S_\nu(1-x).</math>
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| They are related to the [[Legendre chi function]] <math>\chi_\nu</math> as
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| :<math>C_\nu(x) = \mbox{Re} \chi_\nu (e^{ix})</math>
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| and
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| :<math>S_\nu(x) = \mbox{Im} \chi_\nu (e^{ix}).</math>
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| ==Inversion==
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| The Bernoulli and Euler polynomials may be inverted to express the [[monomial]] in terms of the polynomials.
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| Specifically, evidently from the above section on [[#Representation by an integral operator]], it follows that
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| :<math>x^n = \frac {1}{n+1}
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| \sum_{k=0}^n {n+1 \choose k} B_k (x)
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| </math>
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| and
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| :<math>x^n = E_n (x) + \frac {1}{2}
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| \sum_{k=0}^{n-1} {n \choose k} E_k (x).
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| </math>
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| ==Relation to falling factorial==
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| The Bernoulli polynomials may be expanded in terms of the [[falling factorial]] <math>(x)_k</math> as
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| :<math>B_{n+1}(x) = B_{n+1} + \sum_{k=0}^n
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| \frac{n+1}{k+1}
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| \left\{ \begin{matrix} n \\ k \end{matrix} \right\}
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| (x)_{k+1} </math>
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| where <math>B_n=B_n(0)</math> and
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| :<math>\left\{ \begin{matrix} n \\ k \end{matrix} \right\} = S(n,k)</math>
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| denotes the [[Stirling number of the second kind]]. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:
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| :<math>(x)_{n+1} = \sum_{k=0}^n
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| \frac{n+1}{k+1}
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| \left[ \begin{matrix} n \\ k \end{matrix} \right]
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| \left(B_{k+1}(x) - B_{k+1} \right) </math>
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| where
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| :<math>\left[ \begin{matrix} n \\ k \end{matrix} \right] = s(n,k)</math>
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| denotes the [[Stirling number of the first kind]].
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| ==Multiplication theorems==
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| The [[multiplication theorem]]s were given by [[Joseph Ludwig Raabe]] in 1851: | |
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| :<math>B_n(mx)= m^{n-1} \sum_{k=0}^{m-1} B_n \left(x+\frac{k}{m}\right)</math>
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| :<math>E_n(mx)= m^n \sum_{k=0}^{m-1}
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| (-1)^k E_n \left(x+\frac{k}{m}\right)
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| \quad \mbox{ for } m=1,3,\dots</math>
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| :<math>E_n(mx)= \frac{-2}{n+1} m^n \sum_{k=0}^{m-1}
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| (-1)^k B_{n+1} \left(x+\frac{k}{m}\right)
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| \quad \mbox{ for } m=2,4,\dots</math>
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| ==Integrals==
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| Indefinite integrals
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| :<math>\int_a^x B_n(t)\,dt =
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| \frac{B_{n+1}(x)-B_{n+1}(a)}{n+1}</math>
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| :<math>\int_a^x E_n(t)\,dt =
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| \frac{E_{n+1}(x)-E_{n+1}(a)}{n+1}</math>
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| Definite integrals
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| :<math>\int_0^1 B_n(t) B_m(t)\,dt =
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| (-1)^{n-1} \frac{m! n!}{(m+n)!} B_{n+m}
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| \quad \mbox { for } m,n \ge 1 </math>
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| :<math>\int_0^1 E_n(t) E_m(t)\,dt =
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| (-1)^{n} 4 (2^{m+n+2}-1)\frac{m! n!}{(m+n+2)!} B_{n+m+2}</math>
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| ==Periodic Bernoulli polynomials==
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| A '''periodic Bernoulli polynomial''' ''P''<sub>''n''</sub>(''x'') is a Bernoulli polynomial evaluated at the [[fractional part]] 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 wave|sawtooth function]].
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| ==References==
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| <references />
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| * Milton Abramowitz and Irene A. Stegun, eds. ''[[Abramowitz and Stegun|Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables'', (1972) Dover, New York. ''(See [http://www.math.sfu.ca/~cbm/aands/page_804.htm Chapter 23])''
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| * {{Apostol IANT}} ''(See chapter 12.11)''
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| *{{dlmf|first=K. |last=Dilcher|id=24|title=Bernoulli and Euler Polynomials}}
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| * {{Cite journal | last1 = Cvijović | first1 = Djurdje | last2 = Klinowski | first2 = Jacek | year = 1995 | title = New formulae for the Bernoulli and Euler polynomials at rational arguments | url = | journal = Proceedings of the American Mathematical Society | volume = 123 | issue = | pages = 1527–1535 }}
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| * {{Cite journal | doi = 10.1007/s11139-007-9102-0 | last1 = Guillera | first1 = Jesus | last2 = Sondow | first2 = Jonathan | year = 2008 | title = Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent | arxiv = math.NT/0506319 | journal = The Ramanujan Journal | volume = 16 | issue = 3| pages = 247–270 }} ''(Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)''
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| * {{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) | coauthors=[[Robert Charles Vaughan (mathematician)|Robert C. Vaughan]] | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=0-521-84903-9 | pages=495–519 | publisher=Cambridge Univ. Press | location=Cambridge }}
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| [[Category:Special functions]]
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| [[Category:Number theory]]
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| [[Category:Polynomials]]
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