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Bernoulli polynomials
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<h2 id="representations">Representations</h2> <p>The Bernoulli polynomials <i>B</i><i>n</i> can be defined by a <a href="/facts/Generating_function/K7UVjnjL">generating function</a>. They also admit a variety of derived representations. </p> <h3>Generating functions</h3> <p>The generating function for the Bernoulli polynomials is t e x t e t − 1 = ∑ n = 0 ∞ B n ( x ) t n n ! . {\displaystyle {\frac {te^{xt}}{e^{t}-1}}=\sum _{n=0}^{\infty }B_{n}(x){\frac {t^{n}}{n!}}.} The generating function for the Euler polynomials is 2 e x t e t + 1 = ∑ n = 0 ∞ E n ( x ) t n n ! . {\displaystyle {\frac {2e^{xt}}{e^{t}+1}}=\sum _{n=0}^{\infty }E_{n}(x){\frac {t^{n}}{n!}}.} </p> <h3>Explicit formula</h3> <p> B n ( x ) = ∑ k = 0 n ( n k ) B n − k x k , {\displaystyle B_{n}(x)=\sum _{k=0}^{n}{n \choose k}B_{n-k}x^{k},} E m ( x ) = ∑ k = 0 m ( m k ) E k 2 k ( x − 1 2 ) m − k . {\displaystyle E_{m}(x)=\sum _{k=0}^{m}{m \choose k}{\frac {E_{k}}{2^{k}}}\left(x-{\tfrac {1}{2}}\right)^{m-k}.} for n ≥ 0 {\displaystyle n\geq 0} , where B k {\displaystyle B_{k}} are the <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli numbers</a>, and E k {\displaystyle E_{k}} are the <a href="/facts/Euler_numbers/acpKy6uS">Euler numbers</a>. It follows that B n ( 0 ) = B n {\displaystyle B_{n}(0)=B_{n}} and E m ( 1 2 ) = 1 2 m E m {\displaystyle E_{m}{\big (}{\tfrac {1}{2}}{\big )}={\tfrac {1}{2^{m}}}E_{m}} . </p> <h3>Representation by a differential operator</h3> <p>The Bernoulli polynomials are also given by B n ( x ) = D e D − 1 x n {\displaystyle \ B_{n}(x)={\frac {D}{\ e^{D}-1\ }}\ x^{n}\ } where D ≡ d d x {\displaystyle \ D\equiv {\frac {\mathrm {d} }{\ \mathrm {d} x\ }}\ } is differentiation with respect to x and the fraction is expanded as a <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a>. It follows that ∫ a x B n ( u ) d u = B n + 1 ( x ) − B n + 1 ( a ) n + 1 . {\displaystyle \ \int _{a}^{x}\ B_{n}(u)\ \mathrm {d} \ u={\frac {\ B_{n+1}(x)-B_{n+1}(a)\ }{n+1}}~.} cf. § Integrals below. By the same token, the Euler polynomials are given by E n ( x ) = 2 e D + 1 x n . {\displaystyle \ E_{n}(x)={\frac {2}{\ e^{D}+1\ }}\ x^{n}~.} </p> <h3>Representation by an integral operator</h3> <p>The Bernoulli polynomials are also the unique polynomials determined by ∫ x x + 1 B n ( u ) d u = x n . {\displaystyle \int _{x}^{x+1}B_{n}(u)\,du=x^{n}.} </p><p>The <a href="/facts/Integral_transform/AB2yjHJJ">integral transform</a> ( T f ) ( x ) = ∫ x x + 1 f ( u ) d u {\displaystyle (Tf)(x)=\int _{x}^{x+1}f(u)\,du} on polynomials <i>f</i>, simply amounts to ( T f ) ( x ) = e D − 1 D f ( x ) = ∑ n = 0 ∞ D n ( n + 1 ) ! f ( x ) = f ( x ) + f ′ ( x ) 2 + f ″ ( x ) 6 + f ‴ ( x ) 24 + ⋯ . {\displaystyle {\begin{aligned}(Tf)(x)={e^{D}-1 \over D}f(x)&{}=\sum _{n=0}^{\infty }{D^{n} \over (n+1)!}f(x)\\&{}=f(x)+{f'(x) \over 2}+{f''(x) \over 6}+{f'''(x) \over 24}+\cdots .\end{aligned}}} This can be used to produce the inversion formulae below. </p> <h2 id="integral-recurrence">Integral Recurrence</h2> <p>In,<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> it is deduced and proved that the Bernoulli polynomials can be obtained by the following integral recurrence B m ( x ) = m ∫ 0 x B m − 1 ( t ) d t − m ∫ 0 1 ∫ 0 t B m − 1 ( s ) d s d t . {\displaystyle B_{m}(x)=m\int _{0}^{x}B_{m-1}(t)\,dt-m\int _{0}^{1}\int _{0}^{t}B_{m-1}(s)\,dsdt.} </p> <h2 id="another-explicit-formula">Another explicit formula</h2> <p>An explicit formula for the Bernoulli polynomials is given by B n ( x ) = ∑ k = 0 n [ 1 k + 1 ∑ ℓ = 0 k ( − 1 ) ℓ ( k ℓ ) ( x + ℓ ) n ] . {\displaystyle B_{n}(x)=\sum _{k=0}^{n}{\biggl [}{\frac {1}{k+1}}\sum _{\ell =0}^{k}(-1)^{\ell }{k \choose \ell }(x+\ell )^{n}{\biggr ]}.} </p><p>That is similar to the series expression for the <a href="/facts/Hurwitz_zeta_function/4EH7sKXB">Hurwitz zeta function</a> in the complex plane. Indeed, there is the relationship B n ( x ) = − n ζ ( 1 − n , x ) {\displaystyle B_{n}(x)=-n\zeta (1-n,\,x)} where ζ ( s , q ) {\displaystyle \zeta (s,\,q)} is the <a href="/facts/Hurwitz_zeta_function/4EH7sKXB">Hurwitz zeta function</a>. The latter generalizes the Bernoulli polynomials, allowing for non-integer values of n. </p><p>The inner sum may be understood to be the nth <a href="/facts/Forward_difference/aY1txFhj">forward difference</a> of x m , {\displaystyle x^{m},} that is, Δ n x m = ∑ k = 0 n ( − 1 ) n − k ( n k ) ( x + k ) m {\displaystyle \Delta ^{n}x^{m}=\sum _{k=0}^{n}(-1)^{n-k}{n \choose k}(x+k)^{m}} where Δ {\displaystyle \Delta } is the <a href="/facts/Forward_difference_operator/aY1txFhj">forward difference operator</a>. Thus, one may write B n ( x ) = ∑ k = 0 n ( − 1 ) k k + 1 Δ k x n . {\displaystyle B_{n}(x)=\sum _{k=0}^{n}{\frac {(-1)^{k}}{k+1}}\Delta ^{k}x^{n}.} </p><p>This formula may be derived from an identity appearing above as follows. Since the forward difference operator Δ equals Δ = e D − 1 {\displaystyle \Delta =e^{D}-1} where D is differentiation with respect to x, we have, from the <a href="/facts/Mercator_series/BYxgSMVA">Mercator series</a>, D e D − 1 = log ( Δ + 1 ) Δ = ∑ n = 0 ∞ ( − Δ ) n n + 1 . {\displaystyle {\frac {D}{e^{D}-1}}={\frac {\log(\Delta +1)}{\Delta }}=\sum _{n=0}^{\infty }{\frac {(-\Delta )^{n}}{n+1}}.} </p><p>As long as this operates on an mth-degree polynomial such as x m , {\displaystyle x^{m},} one may let n go from 0 only up to m. </p><p>An integral representation for the Bernoulli polynomials is given by the <a href="/facts/N%25C3%25B6rlund%25E2%2580%2593Rice_integral/2WqP5s90">Nörlund–Rice integral</a>, which follows from the expression as a finite difference. </p><p>An explicit formula for the Euler polynomials is given by E n ( x ) = ∑ k = 0 n [ 1 2 k ∑ ℓ = 0 n ( − 1 ) ℓ ( k ℓ ) ( x + ℓ ) n ] . {\displaystyle E_{n}(x)=\sum _{k=0}^{n}\left[{\frac {1}{2^{k}}}\sum _{\ell =0}^{n}(-1)^{\ell }{k \choose \ell }(x+\ell )^{n}\right].} </p><p>The above follows analogously, using the fact that 2 e D + 1 = 1 1 + 1 2 Δ = ∑ n = 0 ∞ ( − 1 2 Δ ) n . {\displaystyle {\frac {2}{e^{D}+1}}={\frac {1}{1+{\tfrac {1}{2}}\Delta }}=\sum _{n=0}^{\infty }{\bigl (}{-{\tfrac {1}{2}}}\Delta {\bigr )}^{n}.} </p> <h2 id="sums-of-pth-powers">Sums of <i>p</i>th powers</h2> <p class="note">Main article: <a href="/facts/Faulhaber%2527s_formula/bQJhbEnp">Faulhaber's formula</a></p> <p>Using either the above integral representation of x n {\displaystyle x^{n}} or the identity B n ( x + 1 ) − B n ( x ) = n x n − 1 {\displaystyle B_{n}(x+1)-B_{n}(x)=nx^{n-1}} , we have ∑ k = 0 x k p = ∫ 0 x + 1 B p ( t ) d t = B p + 1 ( x + 1 ) − B p + 1 p + 1 {\displaystyle \sum _{k=0}^{x}k^{p}=\int _{0}^{x+1}B_{p}(t)\,dt={\frac {B_{p+1}(x+1)-B_{p+1}}{p+1}}} (assuming 00 = 1). </p> <h2 id="explicit-expressions-for-low-degrees">Explicit expressions for low degrees</h2> <p>The first few Bernoulli polynomials are: B 0 ( x ) = 1 , B 4 ( x ) = x 4 − 2 x 3 + x 2 − 1 30 , B 1 ( x ) = x − 1 2 , B 5 ( x ) = x 5 − 5 2 x 4 + 5 3 x 3 − 1 6 x , B 2 ( x ) = x 2 − x + 1 6 , B 6 ( x ) = x 6 − 3 x 5 + 5 2 x 4 − 1 2 x 2 + 1 42 , B 3 ( x ) = x 3 − 3 2 x 2 + 1 2 x | , ⋮ {\displaystyle {\begin{aligned}B_{0}(x)&=1,&B_{4}(x)&=x^{4}-2x^{3}+x^{2}-{\tfrac {1}{30}},\\[4mu]B_{1}(x)&=x-{\tfrac {1}{2}},&B_{5}(x)&=x^{5}-{\tfrac {5}{2}}x^{4}+{\tfrac {5}{3}}x^{3}-{\tfrac {1}{6}}x,\\[4mu]B_{2}(x)&=x^{2}-x+{\tfrac {1}{6}},&B_{6}(x)&=x^{6}-3x^{5}+{\tfrac {5}{2}}x^{4}-{\tfrac {1}{2}}x^{2}+{\tfrac {1}{42}},\\[-2mu]B_{3}(x)&=x^{3}-{\tfrac {3}{2}}x^{2}+{\tfrac {1}{2}}x{\vphantom {\Big |}},\qquad &&\ \,\,\vdots \end{aligned}}} </p><p>The first few Euler polynomials are: E 0 ( x ) = 1 , E 4 ( x ) = x 4 − 2 x 3 + x , E 1 ( x ) = x − 1 2 , E 5 ( x ) = x 5 − 5 2 x 4 + 5 2 x 2 − 1 2 , E 2 ( x ) = x 2 − x , E 6 ( x ) = x 6 − 3 x 5 + 5 x 3 − 3 x , E 3 ( x ) = x 3 − 3 2 x 2 + 1 4 , ⋮ {\displaystyle {\begin{aligned}E_{0}(x)&=1,&E_{4}(x)&=x^{4}-2x^{3}+x,\\[4mu]E_{1}(x)&=x-{\tfrac {1}{2}},&E_{5}(x)&=x^{5}-{\tfrac {5}{2}}x^{4}+{\tfrac {5}{2}}x^{2}-{\tfrac {1}{2}},\\[4mu]E_{2}(x)&=x^{2}-x,&E_{6}(x)&=x^{6}-3x^{5}+5x^{3}-3x,\\[-1mu]E_{3}(x)&=x^{3}-{\tfrac {3}{2}}x^{2}+{\tfrac {1}{4}},\qquad \ \ &&\ \,\,\vdots \end{aligned}}} </p> <h2 id="maximum-and-minimum">Maximum and minimum</h2> <p>At higher n the amount of variation in B n ( x ) {\displaystyle B_{n}(x)} between x = 0 {\displaystyle x=0} and x = 1 {\displaystyle x=1} gets large. For instance, B 16 ( 0 ) = B 16 ( 1 ) = {\displaystyle B_{16}(0)=B_{16}(1)={}} − 3617 510 ≈ − 7.09 , {\displaystyle -{\tfrac {3617}{510}}\approx -7.09,} but B 16 ( 1 2 ) = {\displaystyle B_{16}{\bigl (}{\tfrac {1}{2}}{\bigr )}={}} 118518239 3342336 ≈ 7.09. {\displaystyle {\tfrac {118518239}{3342336}}\approx 7.09.} <a href="/facts/D.H._Lehmer/EI61SICa">Lehmer</a> (1940)<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> showed that the maximum value (Mn) of B n ( x ) {\displaystyle B_{n}(x)} between 0 and 1 obeys M n < 2 n ! ( 2 π ) n {\displaystyle M_{n}<{\frac {2n!}{(2\pi )^{n}}}} unless n is 2 modulo 4, in which case M n = 2 ζ ( n ) n ! ( 2 π ) n {\displaystyle M_{n}={\frac {2\zeta (n)\,n!}{(2\pi )^{n}}}} (where ζ ( x ) {\displaystyle \zeta (x)} is the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>), while the minimum (mn) obeys m n > − 2 n ! ( 2 π ) n {\displaystyle m_{n}>{\frac {-2n!}{(2\pi )^{n}}}} unless <i>n</i> = 0 modulo 4 , in which case m n = − 2 ζ ( n ) n ! ( 2 π ) n . {\displaystyle m_{n}={\frac {-2\zeta (n)\,n!}{(2\pi )^{n}}}.} </p><p>These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well. </p> <h2 id="differences-and-derivatives">Differences and derivatives</h2> <p>The Bernoulli and Euler polynomials obey many relations from <a href="/facts/Umbral_calculus/hN5Ph72G">umbral calculus</a>: Δ B n ( x ) = B n ( x + 1 ) − B n ( x ) = n x n − 1 , Δ E n ( x ) = E n ( x + 1 ) − E n ( x ) = 2 ( x n − E n ( x ) ) . {\displaystyle {\begin{aligned}\Delta B_{n}(x)&=B_{n}(x+1)-B_{n}(x)=nx^{n-1},\\[3mu]\Delta E_{n}(x)&=E_{n}(x+1)-E_{n}(x)=2(x^{n}-E_{n}(x)).\end{aligned}}} (Δ is the <a href="/facts/Forward_difference_operator/aY1txFhj">forward difference operator</a>). Also, E n ( x + 1 ) + E n ( x ) = 2 x n . {\displaystyle E_{n}(x+1)+E_{n}(x)=2x^{n}.} These <a href="/facts/Polynomial_sequence/JaH5aGcH">polynomial sequences</a> are <a href="/facts/Appell_sequence/7hEeFjzM">Appell sequences</a>: B n ′ ( x ) = n B n − 1 ( x ) , E n ′ ( x ) = n E n − 1 ( x ) . {\displaystyle {\begin{aligned}B_{n}'(x)&=nB_{n-1}(x),\\[3mu]E_{n}'(x)&=nE_{n-1}(x).\end{aligned}}} </p> <h3>Translations</h3> <p> B n ( x + y ) = ∑ k = 0 n ( n k ) B k ( x ) y n − k E n ( x + y ) = ∑ k = 0 n ( n k ) E k ( x ) y n − k {\displaystyle {\begin{aligned}B_{n}(x+y)&=\sum _{k=0}^{n}{n \choose k}B_{k}(x)y^{n-k}\\[3mu]E_{n}(x+y)&=\sum _{k=0}^{n}{n \choose k}E_{k}(x)y^{n-k}\end{aligned}}} These identities are also equivalent to saying that these polynomial sequences are <a href="/facts/Appell_sequence/7hEeFjzM">Appell sequences</a>. (<a href="/facts/Hermite_polynomials/MLQNY0ez">Hermite polynomials</a> are another example.) </p> <h3>Symmetries</h3> <p> B n ( 1 − x ) = ( − 1 ) n B n ( x ) , n ≥ 0 , and in particular for n ≠ 1 , B n ( 0 ) = B n ( 1 ) E n ( 1 − x ) = ( − 1 ) n E n ( x ) ( − 1 ) n B n ( − x ) = B n ( x ) + n x n − 1 ( − 1 ) n E n ( − x ) = − E n ( x ) + 2 x n B n ( 1 2 ) = ( 1 2 n − 1 − 1 ) B n , n ≥ 0 from the multiplication theorems below. {\displaystyle {\begin{aligned}B_{n}(1-x)&=\left(-1\right)^{n}B_{n}(x),&&n\geq 0,{\text{ and in particular for }}n\neq 1,~B_{n}(0)=B_{n}(1)\\[3mu]E_{n}(1-x)&=\left(-1\right)^{n}E_{n}(x)\\[1ex]\left(-1\right)^{n}B_{n}(-x)&=B_{n}(x)+nx^{n-1}\\[3mu]\left(-1\right)^{n}E_{n}(-x)&=-E_{n}(x)+2x^{n}\\[1ex]B_{n}{\bigl (}{\tfrac {1}{2}}{\bigr )}&=\left({\frac {1}{2^{n-1}}}-1\right)B_{n},&&n\geq 0{\text{ from the multiplication theorems below.}}\end{aligned}}} <a href="/facts/Zhi-Wei_Sun/Gw78CJDr">Zhi-Wei Sun</a> and Hao Pan <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> established the following surprising symmetry relation: If <i>r</i> + <i>s</i> + <i>t</i> = <i>n</i> and <i>x</i> + <i>y</i> + <i>z</i> = 1, then r [ s , t ; x , y ] n + s [ t , r ; y , z ] n + t [ r , s ; z , x ] n = 0 , {\displaystyle r[s,t;x,y]_{n}+s[t,r;y,z]_{n}+t[r,s;z,x]_{n}=0,} where [ s , t ; x , y ] n = ∑ k = 0 n ( − 1 ) k ( s k ) ( t n − k ) B n − k ( x ) B k ( y ) . {\displaystyle [s,t;x,y]_{n}=\sum _{k=0}^{n}(-1)^{k}{s \choose k}{t \choose {n-k}}B_{n-k}(x)B_{k}(y).} </p> <h2 id="fourier-series">Fourier series</h2> <p>The <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> of the Bernoulli polynomials is also a <a href="/facts/Dirichlet_series/uIAUaLcd">Dirichlet series</a>, given by the expansion B n ( x ) = − n ! ( 2 π i ) n ∑ k ≠ 0 e 2 π i k x k n = − 2 n ! ∑ k = 1 ∞ cos ( 2 k π x − n π 2 ) ( 2 k π ) n . {\displaystyle B_{n}(x)=-{\frac {n!}{(2\pi i)^{n}}}\sum _{k\not =0}{\frac {e^{2\pi ikx}}{k^{n}}}=-2n!\sum _{k=1}^{\infty }{\frac {\cos \left(2k\pi x-{\frac {n\pi }{2}}\right)}{(2k\pi )^{n}}}.} Note the simple large <i>n</i> limit to suitably scaled trigonometric functions. </p><p>This is a special case of the analogous form for the <a href="/facts/Hurwitz_zeta_function/4EH7sKXB">Hurwitz zeta function</a> B n ( x ) = − Γ ( n + 1 ) ∑ k = 1 ∞ exp ( 2 π i k x ) + e i π n exp ( 2 π i k ( 1 − x ) ) ( 2 π i k ) n . {\displaystyle B_{n}(x)=-\Gamma (n+1)\sum _{k=1}^{\infty }{\frac {\exp(2\pi ikx)+e^{i\pi n}\exp(2\pi ik(1-x))}{(2\pi ik)^{n}}}.} </p><p>This expansion is valid only for 0 ≤ <i>x</i> ≤ 1 when <i>n</i> ≥ 2 and is valid for 0 < <i>x</i> < 1 when <i>n</i> = 1. </p><p>The Fourier series of the Euler polynomials may also be calculated. Defining the functions C ν ( x ) = ∑ k = 0 ∞ cos ( ( 2 k + 1 ) π x ) ( 2 k + 1 ) ν S ν ( x ) = ∑ k = 0 ∞ sin ( ( 2 k + 1 ) π x ) ( 2 k + 1 ) ν {\displaystyle {\begin{aligned}C_{\nu }(x)&=\sum _{k=0}^{\infty }{\frac {\cos((2k+1)\pi x)}{(2k+1)^{\nu }}}\\[3mu]S_{\nu }(x)&=\sum _{k=0}^{\infty }{\frac {\sin((2k+1)\pi x)}{(2k+1)^{\nu }}}\end{aligned}}} for ν > 1 {\displaystyle \nu >1} , the Euler polynomial has the Fourier series C 2 n ( x ) = ( − 1 ) n 4 ( 2 n − 1 ) ! π 2 n E 2 n − 1 ( x ) S 2 n + 1 ( x ) = ( − 1 ) n 4 ( 2 n ) ! π 2 n + 1 E 2 n ( x ) . {\displaystyle {\begin{aligned}C_{2n}(x)&={\frac {\left(-1\right)^{n}}{4(2n-1)!}}\pi ^{2n}E_{2n-1}(x)\\[1ex]S_{2n+1}(x)&={\frac {\left(-1\right)^{n}}{4(2n)!}}\pi ^{2n+1}E_{2n}(x).\end{aligned}}} Note that the C ν {\displaystyle C_{\nu }} and S ν {\displaystyle S_{\nu }} are odd and even, respectively: C ν ( x ) = − C ν ( 1 − x ) S ν ( x ) = S ν ( 1 − x ) . {\displaystyle {\begin{aligned}C_{\nu }(x)&=-C_{\nu }(1-x)\\S_{\nu }(x)&=S_{\nu }(1-x).\end{aligned}}} </p><p>They are related to the <a href="/facts/Legendre_chi_function/QF7pSJNt">Legendre chi function</a> χ ν {\displaystyle \chi _{\nu }} as C ν ( x ) = Re χ ν ( e i x ) S ν ( x ) = Im χ ν ( e i x ) . {\displaystyle {\begin{aligned}C_{\nu }(x)&=\operatorname {Re} \chi _{\nu }(e^{ix})\\S_{\nu }(x)&=\operatorname {Im} \chi _{\nu }(e^{ix}).\end{aligned}}} </p> <h2 id="inversion">Inversion</h2> <p>The Bernoulli and Euler polynomials may be inverted to express the <a href="/facts/Monomial/6VAPCYXJ">monomial</a> in terms of the polynomials. </p><p>Specifically, evidently from the above section on integral operators, it follows that x n = 1 n + 1 ∑ k = 0 n ( n + 1 k ) B k ( x ) {\displaystyle x^{n}={\frac {1}{n+1}}\sum _{k=0}^{n}{n+1 \choose k}B_{k}(x)} and x n = E n ( x ) + 1 2 ∑ k = 0 n − 1 ( n k ) E k ( x ) . {\displaystyle x^{n}=E_{n}(x)+{\frac {1}{2}}\sum _{k=0}^{n-1}{n \choose k}E_{k}(x).} </p> <h2 id="relation-to-falling-factorial">Relation to falling factorial</h2> <p>The Bernoulli polynomials may be expanded in terms of the <a href="/facts/Falling_factorial/sAxnmGoC">falling factorial</a> ( x ) k {\displaystyle (x)_{k}} as B n + 1 ( x ) = B n + 1 + ∑ k = 0 n n + 1 k + 1 { n k } ( x ) k + 1 {\displaystyle B_{n+1}(x)=B_{n+1}+\sum _{k=0}^{n}{\frac {n+1}{k+1}}\left\{{\begin{matrix}n\\k\end{matrix}}\right\}(x)_{k+1}} where B n = B n ( 0 ) {\displaystyle B_{n}=B_{n}(0)} and { n k } = S ( n , k ) {\displaystyle \left\{{\begin{matrix}n\\k\end{matrix}}\right\}=S(n,k)} denotes the <a href="/facts/Stirling_number_of_the_second_kind/qg5UrHmk">Stirling number of the second kind</a>. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials: ( x ) n + 1 = ∑ k = 0 n n + 1 k + 1 [ n k ] ( B k + 1 ( x ) − B k + 1 ) {\displaystyle (x)_{n+1}=\sum _{k=0}^{n}{\frac {n+1}{k+1}}\left[{\begin{matrix}n\\k\end{matrix}}\right]\left(B_{k+1}(x)-B_{k+1}\right)} where [ n k ] = s ( n , k ) {\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]=s(n,k)} denotes the <a href="/facts/Stirling_number_of_the_first_kind/Twdk4kA5">Stirling number of the first kind</a>. </p> <h2 id="multiplication-theorems">Multiplication theorems</h2> <p>The <a href="/facts/Multiplication_theorem/I2l5TzmD">multiplication theorems</a> were given by <a href="/facts/Joseph_Ludwig_Raabe/qEfGIj16">Joseph Ludwig Raabe</a> in 1851: </p><p>For a natural number <i>m</i>≥1, B n ( m x ) = m n − 1 ∑ k = 0 m − 1 B n ( x + k m ) {\displaystyle B_{n}(mx)=m^{n-1}\sum _{k=0}^{m-1}B_{n}{\left(x+{\frac {k}{m}}\right)}} E n ( m x ) = m n ∑ k = 0 m − 1 ( − 1 ) k E n ( x + k m ) for odd m E n ( m x ) = − 2 n + 1 m n ∑ k = 0 m − 1 ( − 1 ) k B n + 1 ( x + k m ) for even m {\displaystyle {\begin{aligned}E_{n}(mx)&=m^{n}\sum _{k=0}^{m-1}\left(-1\right)^{k}E_{n}{\left(x+{\frac {k}{m}}\right)}&{\text{ for odd }}m\\[1ex]E_{n}(mx)&={\frac {-2}{n+1}}m^{n}\sum _{k=0}^{m-1}\left(-1\right)^{k}B_{n+1}{\left(x+{\frac {k}{m}}\right)}&{\text{ for even }}m\end{aligned}}} </p> <h2 id="integrals">Integrals</h2> <p>Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> <ul><li> ∫ 0 1 B n ( t ) B m ( t ) d t = ( − 1 ) n − 1 m ! n ! ( m + n ) ! B n + m for m , n ≥ 1 {\displaystyle \int _{0}^{1}B_{n}(t)B_{m}(t)\,dt=(-1)^{n-1}{\frac {m!\,n!}{(m+n)!}}B_{n+m}\quad {\text{for }}m,n\geq 1} </li> <li> ∫ 0 1 E n ( t ) E m ( t ) d t = ( − 1 ) n 4 ( 2 m + n + 2 − 1 ) m ! n ! ( m + n + 2 ) ! B n + m + 2 {\displaystyle \int _{0}^{1}E_{n}(t)E_{m}(t)\,dt=(-1)^{n}4(2^{m+n+2}-1){\frac {m!\,n!}{(m+n+2)!}}B_{n+m+2}} </li></ul> <p>Another integral formula states<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <ul><li> ∫ 0 1 E n ( x + y ) log ( tan π 2 x ) d x = n ! ∑ k = 1 ⌊ n + 1 2 ⌋ ( − 1 ) k − 1 π 2 k ( 2 − 2 − 2 k ) ζ ( 2 k + 1 ) y n + 1 − 2 k ( n + 1 − 2 k ) ! {\displaystyle \int _{0}^{1}E_{n}\left(x+y\right)\log(\tan {\frac {\pi }{2}}x)\,dx=n!\sum _{k=1}^{\left\lfloor {\frac {n+1}{2}}\right\rfloor }{\frac {(-1)^{k-1}}{\pi ^{2k}}}\left(2-2^{-2k}\right)\zeta (2k+1){\frac {y^{n+1-2k}}{(n+1-2k)!}}} </li></ul> <p>with the special case for y = 0 {\displaystyle y=0} </p> <ul><li> ∫ 0 1 E 2 n − 1 ( x ) log ( tan π 2 x ) d x = ( − 1 ) n − 1 ( 2 n − 1 ) ! π 2 n ( 2 − 2 − 2 n ) ζ ( 2 n + 1 ) {\displaystyle \int _{0}^{1}E_{2n-1}\left(x\right)\log(\tan {\frac {\pi }{2}}x)\,dx={\frac {(-1)^{n-1}(2n-1)!}{\pi ^{2n}}}\left(2-2^{-2n}\right)\zeta (2n+1)} </li> <li> ∫ 0 1 B 2 n − 1 ( x ) log ( tan π 2 x ) d x = ( − 1 ) n − 1 π 2 n 2 2 n − 2 ( 2 n − 1 ) ! ∑ k = 1 n ( 2 2 k + 1 − 1 ) ζ ( 2 k + 1 ) ζ ( 2 n − 2 k ) {\displaystyle \int _{0}^{1}B_{2n-1}\left(x\right)\log(\tan {\frac {\pi }{2}}x)\,dx={\frac {(-1)^{n-1}}{\pi ^{2n}}}{\frac {2^{2n-2}}{(2n-1)!}}\sum _{k=1}^{n}(2^{2k+1}-1)\zeta (2k+1)\zeta (2n-2k)} </li> <li> ∫ 0 1 E 2 n ( x ) log ( tan π 2 x ) d x = ∫ 0 1 B 2 n ( x ) log ( tan π 2 x ) d x = 0 {\displaystyle \int _{0}^{1}E_{2n}\left(x\right)\log(\tan {\frac {\pi }{2}}x)\,dx=\int _{0}^{1}B_{2n}\left(x\right)\log(\tan {\frac {\pi }{2}}x)\,dx=0} </li> <li> ∫ 0 1 B 2 n − 1 ( x ) cot ( π x ) d x = 2 ( 2 n − 1 ) ! ( − 1 ) n − 1 ( 2 π ) 2 n − 1 ζ ( 2 n − 1 ) {\displaystyle \int _{0}^{1}{{{B}_{2n-1}}\left(x\right)\cot \left(\pi x\right)dx}={\frac {2\left(2n-1\right)!}{{{\left(-1\right)}^{n-1}}{{\left(2\pi \right)}^{2n-1}}}}\zeta \left(2n-1\right)} </li></ul> <h2 id="periodic-bernoulli-polynomials">Periodic Bernoulli polynomials</h2> <p>A periodic Bernoulli polynomial <i>P</i><i>n</i>(<i>x</i>) is a Bernoulli polynomial evaluated at the <a href="/facts/Fractional_part/lSU6Ap1q">fractional part</a> of the argument <i>x</i>. These functions are used to provide the <a href="/facts/Remainder_term/yDnlsgIe">remainder term</a> in the <a href="/facts/Euler%25E2%2580%2593Maclaurin_formula/mN6uNCzN">Euler–Maclaurin formula</a> relating sums to integrals. The first polynomial is a <a href="/facts/Sawtooth_wave/3QA8wW0h">sawtooth function</a>. </p><p>Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions, and <i>P</i>0(<i>x</i>) is not even a function, being the derivative of a sawtooth and so a <a href="/facts/Dirac_comb/0tx3xslS">Dirac comb</a>. </p><p>The following properties are of interest, valid for all x {\displaystyle x} : </p> <ul><li> P k ( x ) {\displaystyle P_{k}(x)} is continuous for all k > 1 {\displaystyle k>1} </li> <li> P k ′ ( x ) {\displaystyle P_{k}'(x)} exists and is continuous for k > 2 {\displaystyle k>2} </li> <li> P k ′ ( x ) = k P k − 1 ( x ) {\displaystyle P'_{k}(x)=kP_{k-1}(x)} for k > 2 {\displaystyle k>2} </li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Bernoulli_numbers/RH0mblhs">Bernoulli numbers</a></li> <li><a href="/facts/Bernoulli_polynomials_of_the_second_kind/nu5NvaS1">Bernoulli polynomials of the second kind</a></li> <li><a href="/facts/Stirling_polynomial/XWrKj3G5">Stirling polynomial</a></li> <li><a href="/facts/Polynomials_calculating_sums_of_powers_of_arithmetic_progressions/bQJhbEnp">Polynomials calculating sums of powers of arithmetic progressions</a></li></ul> <ul><li>Milton Abramowitz and Irene A. Stegun, eds. <i><a href="/facts/Abramowitz_and_Stegun/CNGSV3Ed">Handbook of Mathematical Functions</a> with Formulas, Graphs, and Mathematical Tables</i>, (1972) Dover, New York. <i>(See Chapter 23)</i></li> <li><a href="/facts/Tom_M._Apostol/S6vGtJWN">Apostol, Tom M.</a> (1976), <i>Introduction to analytic number theory</i>, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90163-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929">0434929</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0335.10001">0335.10001</a> <i>(See chapter 12.11)</i></li> <li>Dilcher, K. (2010), <a href="http://dlmf.nist.gov/24">"Bernoulli and Euler Polynomials"</a>, in <a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</li> <li>Cvijović, Djurdje; Klinowski, Jacek (1995). <a href="https://doi.org/10.1090%2FS0002-9939-1995-1283544-0">"New formulae for the Bernoulli and Euler polynomials at rational arguments"</a>. <i><a href="/facts/Proceedings_of_the_American_Mathematical_Society/W3wRF3MV">Proceedings of the American Mathematical Society</a></i>. 123 (5): 1527–1535. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0002-9939-1995-1283544-0">10.1090/S0002-9939-1995-1283544-0</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2161144">2161144</a>.</li> <li>Guillera, Jesus; Sondow, Jonathan (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". <i>The Ramanujan Journal</i>. 16 (3): 247–270. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.NT/0506319">math.NT/0506319</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs11139-007-9102-0">10.1007/s11139-007-9102-0</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:14910435">14910435</a>. <i>(Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)</i></li> <li><a href="/facts/Hugh_Montgomery_(mathematician)/k1VI3W69">Hugh L. Montgomery</a>; <a href="/facts/Robert_Charles_Vaughan_(mathematician)/djBcTCVL">Robert C. Vaughan</a> (2007). <i>Multiplicative number theory I. Classical theory</i>. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge: Cambridge Univ. Press. pp. 495–519. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-84903-6.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://dlmf.nist.gov/24.7">A list of integral identities involving Bernoulli polynomials</a> from <a href="/facts/NIST/gr9Fnh85">NIST</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Hurtado Benavides, Miguel Ángel. (2020). De las sumas de potencias a las sucesiones de Appell y su caracterización a través de funcionales. [Tesis de maestría]. Universidad Sergio Arboleda. https://repository.usergioarboleda.edu.co/handle/11232/174 <a href="https://repository.usergioarboleda.edu.co/handle/11232/174" target="_blank">https://repository.usergioarboleda.edu.co/handle/11232/174</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Sergio A. Carrillo; Miguel Hurtado. Appell and Sheffer sequences: on their characterizations through functionals and examples. Comptes Rendus. Mathématique, Tome 359 (2021) no. 2, pp. 205-217. doi : 10.5802/crmath.172. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.172/ <a href="https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.172/" target="_blank">https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.172/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Lehmer, D.H. (1940). "On the maxima and minima of Bernoulli polynomials". American Mathematical Monthly. 47 (8): 533–538. doi:10.1080/00029890.1940.11991015. <a href="/wiki/D.H._Lehmer" target="_blank">/wiki/D.H._Lehmer</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Zhi-Wei Sun; Hao Pan (2006). "Identities concerning Bernoulli and Euler polynomials". Acta Arithmetica. 125 (1): 21–39. arXiv:math/0409035. Bibcode:2006AcAri.125...21S. doi:10.4064/aa125-1-3. S2CID 10841415. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Takashi Agoh & Karl Dilcher (2011). "Integrals of products of Bernoulli polynomials". Journal of Mathematical Analysis and Applications. 381: 10–16. doi:10.1016/j.jmaa.2011.03.061. <a href="https://doi.org/10.1016%2Fj.jmaa.2011.03.061" target="_blank">https://doi.org/10.1016%2Fj.jmaa.2011.03.061</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Elaissaoui, Lahoucine & Guennoun, Zine El Abidine (2017). "Evaluation of log-tangent integrals by series involving ζ(2n+1)". Integral Transforms and Special Functions. 28 (6): 460–475. arXiv:1611.01274. doi:10.1080/10652469.2017.1312366. S2CID 119132354. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>