Bernoulli polynomials#Sums of pth powers

The Bernoulli polynomials B_n can be defined by a generating function. They also admit a variety of derived representations.

The generating function for the Bernoulli polynomials is

$$\backslash frac\{t\; e^\{xt\}\}\{e^t-1\}=\; \backslash sum\_\{n=0\}^\backslash infty\; B\_n(x)\; \backslash frac\{t^n\}\{n!\}.$$

The generating function for the Euler polynomials is

$$\backslash frac\{2\; e^\{xt\}\}\{e^t+1\}=\; \backslash sum\_\{n=0\}^\backslash infty\; E\_n(x)\; \backslash frac\{t^n\}\{n!\}.$$

$$B\_n(x)\; =\; \backslash sum\_\{k=0\}^n\; \{n\; \backslash choose\; k\}\; B\_\{n-k\}\; x^k,$$

$$E\_m(x)=$$

\sum_{k=0}^m {m \choose k} \frac{E_k}{2^k}

\left(x-\tfrac12\right)^{m-k} .

for $n\; \backslash geq\; 0$, where $B\_k$ are the Bernoulli numbers, and $E\_k$ are the Euler numbers. It follows that $B\_n(0)\; =\; B\_n$ and $E\_m\backslash big(\backslash tfrac\{1\}\{2\}\backslash big)\; =\; \backslash tfrac\{1\}\{2^m\}\; E\_m$.

The Bernoulli polynomials are also given by

$$\backslash \; B\_n(x)\; =\; \backslash frac\{\; D\; \}\{\backslash \; e^D\; -1\backslash \; \}\backslash \; x^n\backslash$$

where $\backslash \; D\; \backslash equiv\; \backslash frac\{\; \backslash mathrm\{d\}\; \}\{\backslash \; \backslash mathrm\{d\}\; x\backslash \; \}\backslash$ is differentiation with respect to {{mvar|x}} and the fraction is expanded as a formal power series. It follows that

$$\backslash \; \backslash int\_a^x\backslash \; B\_n(u)\backslash \; \backslash mathrm\{d\}\backslash \; u\; =\; \backslash frac\{\backslash \; B\_\{n+1\}(x)\; -\; B\_\{n+1\}(a)\backslash \; \}\{\; n\; +\; 1\; \}\; ~.$$

cf. {{slink||Integrals}} below. By the same token, the Euler polynomials are given by

$$\backslash \; E\_n(x)\; =\; \backslash frac\{\; 2\; \}\{\backslash \; e^D\; +\; 1\backslash \; \}\backslash \; x^n\; ~.$$

The Bernoulli polynomials are also the unique polynomials determined by

$$\backslash int\_x^\{x+1\}\; B\_n(u)\backslash ,du\; =\; x^n.$$

The integral transform

$$(Tf)(x)\; =\; \backslash int\_x^\{x+1\}\; f(u)\backslash ,du$$

on polynomials f, simply amounts to

$$\backslash begin\{align\}$$

(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{align}

This can be used to produce the inversion formulae below.

In,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/11232/174Sergio 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/ it is deduced and proved that the Bernoulli polynomials can be obtained by the following integral recurrence

$$B\_\{m\}(x)=m\; \backslash int\_\{0\}^\{x\}\; B\_\{m-1\}(t)\backslash ,dt-m\backslash int\_\{0\}^\{1\}\; \backslash int\_0^t\; B\_\{m-1\}(s)\backslash ,ds\; dt.$$

An explicit formula for the Bernoulli polynomials is given by

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].

That is similar to the series expression for the Hurwitz zeta function in the complex plane. Indeed, there is the relationship

$$B\_n(x)\; =\; -n\; \backslash zeta(1\; -\; n,\backslash ,x)$$

where $\backslash zeta(s,\backslash ,q)$ is the Hurwitz zeta function. The latter generalizes the Bernoulli polynomials, allowing for non-integer values {{nobr|of {{mvar|n}}.}}

The inner sum may be understood to be the {{mvar|n}}th forward difference of $x^m,$ that is,

$$\backslash Delta^n\; x^m\; =\; \backslash sum\_\{k=0\}^n\; (-1)^\{n\; -\; k\}\{n\; \backslash choose\; k\}(x\; +\; k)^m$$

where $\backslash Delta$ is the forward difference operator. Thus, one may write

$$B\_n(x)\; =\; \backslash sum\_\{k=0\}^n\; \backslash frac\{(-1)^k\}\{k\; +\; 1\}\backslash Delta^k\; x^n.$$

This formula may be derived from an identity appearing above as follows. Since the forward difference operator {{math|Δ}} equals

$$\backslash Delta\; =\; e^D\; -\; 1$$

where {{mvar|D}} is differentiation with respect to {{mvar|x}}, we have, from the Mercator series,

$$\backslash frac\{\; D\; \}\{e^D\; -\; 1\}\; =\; \backslash frac\{\backslash log(\backslash Delta\; +\; 1)\}\{\backslash Delta\}\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-\backslash Delta)^n\; \}\{n\; +\; 1\}.$$

As long as this operates on an {{mvar|m}}th-degree polynomial such as $x^m,$ one may let {{mvar|n}} go from {{math|0}} only up {{nobr|to {{mvar|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

$$E\_n(x)\; =\; \backslash sum\_\{k=0\}^n\; \backslash left[\; \backslash frac\{1\}\{2^k\}\backslash sum\_\{\backslash ell=0\}^n\; (-1)^\backslash ell\; \{k\; \backslash choose\; \backslash ell\}(x\; +\; \backslash ell)^n\; \backslash right]\; .$$

The above follows analogously, using the fact that

$$\backslash frac\{2\}\{e^D\; +\; 1\}\; =\; \backslash frac\{1\}\{1\; +\; \backslash tfrac12\; \backslash Delta\}\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; \backslash bigl(\; \{-\backslash tfrac\{1\}\{2\}\}\; \backslash Delta\; \backslash bigr)^n\; .$$

{{main|Faulhaber's formula}}

Using either the above integral representation of $x^n$ or the identity $B\_n(x\; +\; 1)\; -\; B\_n(x)\; =\; nx^\{n-1\}$, we have

$$\backslash sum\_\{k=0\}^x\; k^p\; =\; \backslash int\_0^\{x+1\}\; B\_p(t)\; \backslash ,\; dt\; =\; \backslash frac\{B\_\{p+1\}(x+1)-B\_\{p+1\}\}\{p+1\}$$

(assuming 0⁰ = 1).

The first few Bernoulli polynomials are:

\begin{align}

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{align}

The first few Euler polynomials are:

\begin{align}

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{align}

At higher {{mvar|n}} the amount of variation in $B\_n(x)$ between $x\; =\; 0$ and $x\; =\; 1$ gets large. For instance, $B\_\{16\}(0)\; =\; B\_\{16\}(1)\; =\; \{\}$$-\backslash tfrac\{3617\}\{510\}\; \backslash approx\; -7.09,$ but $B\_\{16\}\backslash bigl(\backslash tfrac12\backslash bigr)\; =\; \{\}$$\backslash tfrac\{118518239\}\{3342336\}\; \backslash approx\; 7.09.$ {{nobr|Lehmer (1940){{cite journal |first=D.H. |last=Lehmer |author-link=D.H. Lehmer |year=1940 |title=On the maxima and minima of Bernoulli polynomials |journal=American Mathematical Monthly |volume=47 |issue=8 |pages=533–538 |doi=10.1080/00029890.1940.11991015 }}}} showed that the maximum value ({{mvar|M{{sub|n}}}}) of $B\_n(x)$ between {{math|0}} and {{math|1}} obeys

unless {{mvar|n}} is {{nobr|{{math|2 modulo 4}},}} in which case

$$M\_n\; =\; \backslash frac\{2\backslash zeta\; (n)\backslash ,n!\}\{(2\backslash pi)^n\}$$

(where $\backslash zeta(x)$ is the Riemann zeta function), while the minimum ({{mvar|m{{sub|n}}}}) obeys

$$m\_n\; >\; \backslash frac\{\; -2\; n!\}\{(2\backslash pi)^n\}$$

unless {{nobr| {{math|1=n = 0 modulo 4 }} ,}} in which case

$$m\_n\; =\; \backslash frac\{-2\; \backslash zeta(n)\backslash ,n!\; \}\{(2\backslash pi)^n\}.$$

These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well.

The Bernoulli and Euler polynomials obey many relations from umbral calculus:

$$\backslash begin\{align\}$$

\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{align}

({{math|Δ}} is the forward difference operator). Also,

$$E\_n(x+1)\; +\; E\_n(x)\; =\; 2x^n.$$

These polynomial sequences are Appell sequences:

$$\backslash begin\{align\}$$

B_n'(x) &= n B_{n-1}(x), \\[3mu]

E_n'(x) &= n E_{n-1}(x).

\end{align}

$$\backslash begin\{align\}$$

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{align}

These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials are another example.)

$$\backslash begin\{align\}$$

B_n(1-x) &= \left(-1\right)^n B_n(x), && n \ge 0,

\text{ and in particular for } n \ne 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(\tfrac12\bigr) &= \left(\frac{1}{2^{n-1}}-1\right) B_n, && n \geq 0\text{ from the multiplication theorems below.}

\end{align}

Zhi-Wei Sun and Hao Pan {{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 |issue=1 |arxiv=math/0409035 |doi=10.4064/aa125-1-3|bibcode=2006AcAri.125...21S |s2cid=10841415 }} established the following surprising symmetry relation: If {{math|1= r + s + t = n}} and {{math|1= x + y + z = 1}}, then

$$r[s,t;x,y]\_n+s[t,r;y,z]\_n+t[r,s;z,x]\_n=0,$$

where

$$[s,t;x,y]\_n=\backslash sum\_\{k=0\}^n(-1)^k\{s\; \backslash choose\; k\}\{t\backslash choose\; \{n-k\}\}\; B\_\{n-k\}(x)B\_k(y).$$

The Fourier series of the Bernoulli polynomials is also a Dirichlet series, given by the expansion

$$B\_n(x)\; =\; -\backslash frac\{n!\}\{(2\backslash pi\; i)^n\}\backslash sum\_\{k\backslash not=0\; \}\backslash frac\{e^\{2\backslash pi\; ikx\}\}\{k^n\}=\; -2\; n!\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash frac\{\backslash cos\backslash left(2\; k\; \backslash pi\; x-\; \backslash frac\{n\; \backslash pi\}\; 2\; \backslash right)\}\{(2\; k\; \backslash pi)^n\}.$$

Note the simple large n limit to suitably scaled trigonometric functions.

This is a special case of the analogous form for the Hurwitz zeta function

$$B\_n(x)\; =\; -\backslash Gamma(n+1)\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{\; \backslash exp\; (2\backslash pi\; ikx)\; +\; e^\{i\backslash pi\; n\}\; \backslash exp\; (2\backslash pi\; ik(1-x))\; \}\; \{\; (2\backslash pi\; ik)^n\; \}.$$

This expansion is valid only for {{math|0 ≤ x ≤ 1}} when {{math|n ≥ 2}} and is valid for {{math|0 < x < 1}} when {{math|1=n = 1}}.

The Fourier series of the Euler polynomials may also be calculated. Defining the functions

$$\backslash begin\{align\}$$

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{align}

for $\backslash nu\; >\; 1$, the Euler polynomial has the Fourier series

$$\backslash begin\{align\}$$

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{align}

Note that the $C\_\backslash nu$ and $S\_\backslash nu$ are odd and even, respectively:$$\backslash begin\{align\}$$

C_\nu(x) &= -C_\nu(1-x) \\

S_\nu(x) &= S_\nu(1-x).

\end{align}

They are related to the Legendre chi function $\backslash chi\_\backslash nu$ as

$$\backslash begin\{align\}$$

C_\nu(x) &= \operatorname{Re} \chi_\nu (e^{ix}) \\

S_\nu(x) &= \operatorname{Im} \chi_\nu (e^{ix}).

\end{align}

The Bernoulli and Euler polynomials may be inverted to express the monomial in terms of the polynomials.

Specifically, evidently from the above section on integral operators, it follows that

$$x^n\; =\; \backslash frac\; \{1\}\{n+1\}\; \backslash sum\_\{k=0\}^n\; \{n+1\; \backslash choose\; k\}\; B\_k\; (x)$$

and

$$x^n\; =\; E\_n\; (x)\; +\; \backslash frac\; \{1\}\{2\}\; \backslash sum\_\{k=0\}^\{n-1\}\; \{n\; \backslash choose\; k\}\; E\_k\; (x).$$

The Bernoulli polynomials may be expanded in terms of the falling factorial $(x)\_k$ as

$$B\_\{n+1\}(x)\; =\; B\_\{n+1\}\; +\; \backslash sum\_\{k=0\}^n$$

\frac{n+1}{k+1}

\left\{ \begin{matrix} n \\ k \end{matrix} \right\}

(x)_{k+1}

where $B\_n\; =\; B\_n(0)$ and

$$\backslash left\backslash \{\; \backslash begin\{matrix\}\; n\; \backslash \backslash \; k\; \backslash end\{matrix\}\; \backslash right\backslash \}\; =\; S(n,k)$$

denotes the Stirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:

$$(x)\_\{n+1\}\; =\; \backslash sum\_\{k=0\}^n$$

\frac{n+1}{k+1}

\left[ \begin{matrix} n \\ k \end{matrix} \right]

\left(B_{k+1}(x) - B_{k+1} \right)

where

$$\backslash left[\; \backslash begin\{matrix\}\; n\; \backslash \backslash \; k\; \backslash end\{matrix\}\; \backslash right]\; =\; s(n,k)$$

denotes the Stirling number of the first kind.

The multiplication theorems were given by Joseph Ludwig Raabe in 1851:

For a natural number {{math|m≥1}},

$$B\_n(mx)=\; m^\{n-1\}\; \backslash sum\_\{k=0\}^\{m-1\}\; B\_n\{\backslash left(x+\backslash frac\{k\}\{m\}\backslash right)\}$$

$$\backslash begin\{align\}$$

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{align}

Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:{{cite journal |name-list-style=amp |author1=Takashi Agoh |author2=Karl Dilcher |journal=Journal of Mathematical Analysis and Applications |volume=381 |year=2011 |pages=10–16 |title=Integrals of products of Bernoulli polynomials | doi=10.1016/j.jmaa.2011.03.061 |doi-access=free }}

• $\backslash int\_0^1\; B\_n(t)\; B\_m(t)\backslash ,dt\; =\; (-1)^\{n-1\}\; \backslash frac\{m!\backslash ,\; n!\}\{(m+n)!\}\; B\_\{n+m\}\; \backslash quad\; \backslash text\{for\; \}\; m,n\; \backslash geq\; 1$
• $\backslash int\_0^1\; E\_n(t)\; E\_m(t)\backslash ,dt\; =\; (-1)^\{n\}\; 4\; (2^\{m+n+2\}-1)\backslash frac\{m!\backslash ,n!\}\{(m+n+2)!\}\; B\_\{n+m+2\}$

Another integral formula states{{cite journal | author=Elaissaoui, Lahoucine | author2=Guennoun, Zine El Abidine | name-list-style=amp | title=Evaluation of log-tangent integrals by series involving ζ(2n+1)| journal=Integral Transforms and Special Functions | language=English | year=2017| volume=28 | issue=6 | pages=460–475 | doi=10.1080/10652469.2017.1312366 | arxiv=1611.01274 | s2cid=119132354 }}

• $\backslash int\_0^\{1\}E\_\{n\}\backslash left(\; x\; +y\backslash right)\backslash log(\backslash tan\; \backslash frac\{\backslash pi\}\{2\}x)\backslash ,dx=\; n!\; \backslash sum\_\{k=1\}^\{\backslash left\backslash lfloor\backslash frac\; \{n+1\}2\backslash right\backslash rfloor\}\; \backslash frac\{(-1)^\{k-1\}\}\{\; \backslash pi^\{2k\}\}\; \backslash left(\; 2-2^\{-2k\}\; \backslash right)\backslash zeta(2k+1)\; \backslash frac\{y^\; \{n+1-2k\}\}\{(n\; +1-\; 2k)!\}$

with the special case for $y=0$

• $\backslash int\_0^\{1\}E\_\{2n-1\}\backslash left(\; x\; \backslash right)\backslash log(\backslash tan\; \backslash frac\{\backslash pi\}\{2\}x)\backslash ,dx=$

\frac{(-1)^{n-1}(2n-1)!}{\pi^{2n}}\left( 2-2^{-2n} \right)\zeta(2n+1)

• $\backslash int\_0^\{1\}B\_\{2n-1\}\backslash left(\; x\; \backslash right)\backslash log(\backslash tan\; \backslash frac\{\backslash pi\}\{2\}x)\backslash ,dx=$

\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)

• $\backslash int\_0^\{1\}E\_\{2n\}\backslash left(\; x\; \backslash right)\backslash log(\backslash tan\; \backslash frac\{\backslash pi\}\{2\}x)\backslash ,dx=\backslash int\_0^\{1\}B\_\{2n\}\backslash left(\; x\; \backslash right)\backslash log(\backslash tan\; \backslash frac\{\backslash pi\}\{2\}x)\backslash ,dx=0$
• $\backslash int\_\{0\}^\{1\}\{\{\{B\}\_\{2n-1\}\}\backslash left(\; x\; \backslash right)\backslash cot\; \backslash left(\; \backslash pi\; x\; \backslash right)dx\}=\backslash frac\{2\backslash left(\; 2n-1\; \backslash right)!\}\{\{\{\backslash left(\; -1\; \backslash right)\}^\{n-1\}\}\{\{\backslash left(\; 2\backslash pi\; \backslash right)\}^\{2n-1\}\}\}\backslash zeta\; \backslash left(\; 2n-1\; \backslash right)$

A periodic Bernoulli polynomial {{math|P_n(x)}} is a Bernoulli polynomial evaluated at the fractional part of the argument {{math|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.

Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions, and {{math|P₀(x)}} is not even a function, being the derivative of a sawtooth and so a Dirac comb.

The following properties are of interest, valid for all $x$:

• $P\_k(x)$ is continuous for all $k\; >\; 1$
• $P\_k\text{'}(x)$ exists and is continuous for $k\; >\; 2$
• $P\text{'}\_k(x)\; =\; k\; P\_\{k-1\}(x)$ for $k\; >\; 2$

• Bernoulli numbers
• Bernoulli polynomials of the second kind
• Stirling polynomial
• Polynomials calculating sums of powers of arithmetic progressions

{{reflist}}

{{refbegin}}

• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (1972) Dover, New York. (See Chapter 23)
• {{Apostol IANT}} (See chapter 12.11)
• {{dlmf|first=K. |last=Dilcher|id=24|title=Bernoulli and Euler Polynomials}}
• {{Cite journal | last1 = Cvijović | first1 = Djurdje | last2 = Klinowski | first2 = Jacek | year = 1995 | title = New formulae for the Bernoulli and Euler polynomials at rational arguments | journal = Proceedings of the American Mathematical Society | volume = 123 | issue = 5 | pages = 1527–1535 | doi=10.1090/S0002-9939-1995-1283544-0 | doi-access = free | jstor = 2161144 }}
• {{cite arXiv |first=Omran |last=Kouba |title=Lecture Notes, Bernoulli Polynomials and Applications |date=2016 |class=math.CA |eprint=1309.7560v2}}
• {{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 | s2cid = 14910435 }} (Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)
• {{cite book | author=Hugh L. Montgomery | author-link=Hugh Montgomery (mathematician) |author2=Robert C. Vaughan |author-link2=Robert Charles Vaughan (mathematician) | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=978-0-521-84903-6 | pages=495–519 | publisher=Cambridge Univ. Press | location=Cambridge }}

{{refend}}

• [https://dlmf.nist.gov/24.7 A list of integral identities involving Bernoulli polynomials] from NIST

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Category:Special functions

Category:Number theory

Category:Polynomials