§24.4 Basic Properties

Permalink:: http://dlmf.nist.gov/24.4

§24.4(i) Difference Equations
§24.4(ii) Symmetry
§24.4(iii) Sums of Powers
§24.4(iv) Finite Expansions
§24.4(v) Multiplication Formulas
§24.4(vi) Special Values
§24.4(vii) Derivatives
§24.4(viii) Symbolic Operations
§24.4(ix) Relations to Other Functions

§24.4(i) Difference Equations

Notes:: See Nörlund (1924, pp. 18 and 23) or Milne-Thomson (1933, pp. 136 and 146).
Keywords:: Bernoulli polynomials, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.4.i

24.4.1 $\mathop{B_{{n}}\/}\nolimits\!\left(x+1\right)-\mathop{B_{{n}}\/}\nolimits\!\left(x\right)=nx^{{n-1}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.6
Referenced by:: §24.13(i)
Permalink:: http://dlmf.nist.gov/24.4.E1
Encodings:: TeX, pMML, png

24.4.2 $\mathop{E_{{n}}\/}\nolimits\!\left(x+1\right)+\mathop{E_{{n}}\/}\nolimits\!\left(x\right)=2x^{n}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.6
Permalink:: http://dlmf.nist.gov/24.4.E2
Encodings:: TeX, pMML, png

§24.4(ii) Symmetry

Notes:: See Nörlund (1924, pp. 21 and 24) or Milne-Thomson (1933, pp. 136 and 146).
Keywords:: Bernoulli polynomials, Euler polynomials
Referenced by:: §24.4(vi)
Permalink:: http://dlmf.nist.gov/24.4.ii

24.4.3 $\mathop{B_{{n}}\/}\nolimits\!\left(1-x\right)=(-1)^{n}\mathop{B_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.8
Permalink:: http://dlmf.nist.gov/24.4.E3
Encodings:: TeX, pMML, png

24.4.4 $\mathop{E_{{n}}\/}\nolimits\!\left(1-x\right)=(-1)^{n}\mathop{E_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.8
Permalink:: http://dlmf.nist.gov/24.4.E4
Encodings:: TeX, pMML, png

24.4.5 $(-1)^{n}\mathop{B_{{n}}\/}\nolimits\!\left(-x\right)=\mathop{B_{{n}}\/}\nolimits\!\left(x\right)+nx^{{n-1}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.9
Permalink:: http://dlmf.nist.gov/24.4.E5
Encodings:: TeX, pMML, png

24.4.6 $(-1)^{{n+1}}\mathop{E_{{n}}\/}\nolimits\!\left(-x\right)=\mathop{E_{{n}}\/}\nolimits\!\left(x\right)-2x^{n}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.9
Permalink:: http://dlmf.nist.gov/24.4.E6
Encodings:: TeX, pMML, png

§24.4(iii) Sums of Powers

Notes:: See Nörlund (1924, p. 19) or Milne-Thomson (1933, pp. 137 and 147). For (24.4.9) and (24.4.10) see Howard (1996b) and Slavutskiĭ (2000). For (24.4.11) see Apostol (2006).
Keywords:: Bernoulli polynomials, Euler polynomials, sums of powers
Permalink:: http://dlmf.nist.gov/24.4.iii

24.4.7 $\sum _{{k=1}}^{m}k^{n}=\frac{\mathop{B_{{n+1}}\/}\nolimits\!\left(m+1\right)-\mathop{B_{{n+1}}\/}\nolimits}{n+1},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $k$ : integer, $m$ : integer and $n$ : integer
A&S Ref:: 23.1.4
Referenced by:: §24.17(iii)
Permalink:: http://dlmf.nist.gov/24.4.E7
Encodings:: TeX, pMML, png

24.4.8 $\sum _{{k=1}}^{m}(-1)^{{m-k}}k^{n}=\frac{\mathop{E_{{n}}\/}\nolimits\!\left(m+1\right)+(-1)^{m}\mathop{E_{{n}}\/}\nolimits\!\left(0\right)}{2}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $k$ : integer, $m$ : integer and $n$ : integer
A&S Ref:: 23.1.4
Referenced by:: §24.17(iii)
Permalink:: http://dlmf.nist.gov/24.4.E8
Encodings:: TeX, pMML, png

24.4.9 $\sum _{{k=0}}^{{m-1}}(a+dk)^{n}={\frac{d^{n}}{n+1}\left(\mathop{B_{{n+1}}\/}\nolimits\!\left(m+\frac{a}{d}\right)-\mathop{B_{{n+1}}\/}\nolimits\!\left(\frac{a}{d}\right)\right)},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $k$ : integer, $m$ : integer and $n$ : integer
Referenced by:: §24.4(iii)
Permalink:: http://dlmf.nist.gov/24.4.E9
Encodings:: TeX, pMML, png

24.4.10 $\sum _{{k=0}}^{{m-1}}(-1)^{{k}}(a+dk)^{n}={\frac{d^{n}}{2}\left((-1)^{{m-1}}\mathop{E_{{n}}\/}\nolimits\!\left(m+\frac{a}{d}\right)+\mathop{E_{{n}}\/}\nolimits\!\left(\frac{a}{d}\right)\right)}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $k$ : integer, $m$ : integer and $n$ : integer
Referenced by:: §24.4(iii)
Permalink:: http://dlmf.nist.gov/24.4.E10
Encodings:: TeX, pMML, png

24.4.11 $\sum _{{\substack{k=1\\ (k,m)=1}}}^{m}k^{n}=\frac{1}{n+1}\sum _{{j=1}}^{{n+1}}{n+1\choose j}\*\left(\prod _{{p\divides m}}(1-p^{{n-j}})\mathop{B_{{n+1-j}}\/}\nolimits\right)m^{j}.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $(a,b)$ : open interval, $j$ : integer, $k$ : integer, $m$ : integer, $n$ : integer and $p$ : prime
Referenced by:: §24.4(iii)
Permalink:: http://dlmf.nist.gov/24.4.E11
Encodings:: TeX, pMML, png

§24.4(iv) Finite Expansions

Notes:: These identities follow from (24.2.3) and (24.2.8).
Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.4.iv

24.4.12 $\mathop{B_{{n}}\/}\nolimits\!\left(x+h\right)=\sum _{{k=0}}^{n}{n\choose k}\mathop{B_{{k}}\/}\nolimits\!\left(x\right)h^{{n-k}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.7
Permalink:: http://dlmf.nist.gov/24.4.E12
Encodings:: TeX, pMML, png

24.4.13 $\mathop{E_{{n}}\/}\nolimits\!\left(x+h\right)=\sum _{{k=0}}^{n}{n\choose k}\mathop{E_{{k}}\/}\nolimits\!\left(x\right)h^{{n-k}},$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.7
Permalink:: http://dlmf.nist.gov/24.4.E13
Encodings:: TeX, pMML, png

24.4.14 $\mathop{E_{{n-1}}\/}\nolimits\!\left(x\right)=\frac{2}{n}\sum _{{k=0}}^{n}{n\choose k}(1-2^{k})\mathop{B_{{k}}\/}\nolimits x^{{n-k}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.4.E14
Encodings:: TeX, pMML, png

24.4.15 $\mathop{B_{{2n}}\/}\nolimits=\frac{2n}{2^{{2n}}(2^{{2n}}-1)}\sum _{{k=0}}^{{n-1}}{2n-1\choose 2k}\mathop{E_{{2k}}\/}\nolimits,$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.4.E15
Encodings:: TeX, pMML, png

24.4.16 $\mathop{E_{{2n}}\/}\nolimits=\frac{1}{2n+1}-\sum _{{k=1}}^{n}{2n\choose 2k-1}\frac{2^{{2k}}(2^{{2k-1}}-1)\mathop{B_{{2k}}\/}\nolimits}{k},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.4.E16
Encodings:: TeX, pMML, png

24.4.17 $\mathop{E_{{2n}}\/}\nolimits=1-\sum _{{k=1}}^{n}{2n\choose 2k-1}\frac{2^{{2k}}(2^{{2k}}-1)\mathop{B_{{2k}}\/}\nolimits}{2k}.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.4.E17
Encodings:: TeX, pMML, png

§24.4(v) Multiplication Formulas

Notes:: See Nörlund (1924, pp. 21 and 24) or Milne-Thomson (1933, p. 141). For (24.4.24) see Todorov (1991).
Keywords:: Bernoulli polynomials
Referenced by:: §24.4(vi)
Permalink:: http://dlmf.nist.gov/24.4.v

¶ Raabe’s Theorem

Keywords:: Bernoulli polynomials, Euler polynomials, Raabe’s theorem

24.4.18 $\mathop{B_{{n}}\/}\nolimits\!\left(mx\right)=m^{{n-1}}\sum _{{k=0}}^{{m-1}}\mathop{B_{{n}}\/}\nolimits\!\left(x+\frac{k}{m}\right).$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $k$ : integer, $m$ : integer, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.10
Permalink:: http://dlmf.nist.gov/24.4.E18
Encodings:: TeX, pMML, png

Next,

24.4.19 $\mathop{E_{{n}}\/}\nolimits\!\left(mx\right)=-\frac{2m^{n}}{n+1}\sum _{{k=0}}^{{m-1}}(-1)^{k}\mathop{B_{{n+1}}\/}\nolimits\!\left(x+\frac{k}{m}\right),$ $m=2,4,6,\dots$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $k$ : integer, $m$ : integer, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.10
Permalink:: http://dlmf.nist.gov/24.4.E19
Encodings:: TeX, pMML, png

24.4.20 $\mathop{E_{{n}}\/}\nolimits\!\left(mx\right)=m^{n}\sum _{{k=0}}^{{m-1}}(-1)^{k}\mathop{E_{{n}}\/}\nolimits\!\left(x+\frac{k}{m}\right),$ $m=1,3,5,\dots$ .

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $k$ : integer, $m$ : integer, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.10
Permalink:: http://dlmf.nist.gov/24.4.E20
Encodings:: TeX, pMML, png

24.4.21 $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)=2^{{n-1}}\left(\mathop{B_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}x\right)+\mathop{B_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}x+\tfrac{1}{2}\right)\right),$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.4.E21
Encodings:: TeX, pMML, png

24.4.22 $\mathop{E_{{n-1}}\/}\nolimits\!\left(x\right)=\frac{2}{n}\left(\mathop{B_{{n}}\/}\nolimits\!\left(x\right)-2^{n}\mathop{B_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}x\right)\right),$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.27
Permalink:: http://dlmf.nist.gov/24.4.E22
Encodings:: TeX, pMML, png

24.4.23 $\mathop{E_{{n-1}}\/}\nolimits\!\left(x\right)=\frac{2^{n}}{n}\left(\mathop{B_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}x+\tfrac{1}{2}\right)-\mathop{B_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}x\right)\right),$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.27
Permalink:: http://dlmf.nist.gov/24.4.E23
Encodings:: TeX, pMML, png

24.4.24 $\mathop{B_{{n}}\/}\nolimits\!\left(mx\right)=m^{n}\mathop{B_{{n}}\/}\nolimits\!\left(x\right)+n\sum _{{k=1}}^{n}\sum _{{j=0}}^{{k-1}}(-1)^{j}{n\choose k}\*\left(\sum _{{r=1}}^{{m-1}}\frac{e^{{2\pi i(k-j)r/m}}}{(1-e^{{2\pi ir/m}})^{n}}\right)(j+mx)^{{n-1}},$ $n=1,2,\dots$ , $m=2,3,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $e$ : base of exponential function, $j$ : integer, $k$ : integer, $m$ : integer, $n$ : integer and $x$ : real or complex
Referenced by:: §24.4(v)
Permalink:: http://dlmf.nist.gov/24.4.E24
Encodings:: TeX, pMML, png

§24.4(vi) Special Values

Notes:: (24.4.25)–(24.4.33) follow from the identities in §§24.4(ii) and 24.4(v).
Keywords:: Bernoulli polynomials, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.4.vi

24.4.25 $\mathop{B_{{n}}\/}\nolimits\!\left(0\right)=(-1)^{n}\mathop{B_{{n}}\/}\nolimits\!\left(1\right)=\mathop{B_{{n}}\/}\nolimits.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.20
Referenced by:: ¶ ‣ §24.17(ii), §24.4(vi)
Permalink:: http://dlmf.nist.gov/24.4.E25
Encodings:: TeX, pMML, png

24.4.26 $\mathop{E_{{n}}\/}\nolimits\!\left(0\right)=-\mathop{E_{{n}}\/}\nolimits\!\left(1\right)=-\frac{2}{n+1}(2^{{n+1}}-1)\mathop{B_{{n+1}}\/}\nolimits,$ $n>0$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials and $n$ : integer
A&S Ref:: 23.1.20
Referenced by:: §24.9, Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/24.4.E26
Encodings:: TeX, pMML, png
Errata (effective with 1.0.5):: This equation is true only for $n>0$ . Previously, $n=0$ was also allowed.
Reported 2012-05-14 by Vladimir Yurovsky

24.4.27 $\mathop{B_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}\right)=-(1-2^{{1-n}})\mathop{B_{{n}}\/}\nolimits,$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.21
Referenced by:: ¶ ‣ §2.10(i), ¶ ‣ §24.17(ii)
Permalink:: http://dlmf.nist.gov/24.4.E27
Encodings:: TeX, pMML, png

24.4.28 $\mathop{E_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}\right)=2^{{-n}}\mathop{E_{{n}}\/}\nolimits.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials and $n$ : integer
A&S Ref:: 23.1.21
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.4.E28
Encodings:: TeX, pMML, png

24.4.29 $\mathop{B_{{2n}}\/}\nolimits\!\left(\tfrac{1}{3}\right)=\mathop{B_{{2n}}\/}\nolimits\!\left(\tfrac{2}{3}\right)=-\tfrac{1}{2}(1-3^{{1-2n}})\mathop{B_{{2n}}\/}\nolimits.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.23
Permalink:: http://dlmf.nist.gov/24.4.E29
Encodings:: TeX, pMML, png

24.4.30 $\mathop{E_{{2n-1}}\/}\nolimits\!\left(\tfrac{1}{3}\right)=-\mathop{E_{{2n-1}}\/}\nolimits\!\left(\tfrac{2}{3}\right)=-\frac{(1-3^{{1-2n}})(2^{{2n}}-1)}{2n}\mathop{B_{{2n}}\/}\nolimits,$ $n=1,2,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials and $n$ : integer
A&S Ref:: 23.1.22
Permalink:: http://dlmf.nist.gov/24.4.E30
Encodings:: TeX, pMML, png

24.4.31 $\mathop{B_{{n}}\/}\nolimits\!\left(\tfrac{1}{4}\right)=(-1)^{n}\mathop{B_{{n}}\/}\nolimits\!\left(\tfrac{3}{4}\right)=-\frac{1-2^{{1-n}}}{2^{n}}\mathop{B_{{n}}\/}\nolimits-\frac{n}{4^{n}}\mathop{E_{{n-1}}\/}\nolimits,$ $n=1,2,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials and $n$ : integer
A&S Ref:: 23.1.22
Permalink:: http://dlmf.nist.gov/24.4.E31
Encodings:: TeX, pMML, png

24.4.32 $\mathop{B_{{2n}}\/}\nolimits\!\left(\tfrac{1}{6}\right)=\mathop{B_{{2n}}\/}\nolimits\!\left(\tfrac{5}{6}\right)=\tfrac{1}{2}(1-2^{{1-2n}})(1-3^{{1-2n}})\mathop{B_{{2n}}\/}\nolimits,$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.24
Permalink:: http://dlmf.nist.gov/24.4.E32
Encodings:: TeX, pMML, png

24.4.33 $\mathop{E_{{2n}}\/}\nolimits\!\left(\tfrac{1}{6}\right)=\mathop{E_{{2n}}\/}\nolimits\!\left(\tfrac{5}{6}\right)=\frac{1+3^{{-2n}}}{2^{{2n+1}}}\mathop{E_{{2n}}\/}\nolimits.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials and $n$ : integer
Referenced by:: §24.4(vi)
Permalink:: http://dlmf.nist.gov/24.4.E33
Encodings:: TeX, pMML, png

§24.4(vii) Derivatives

Notes:: Use (24.2.3) and (24.2.8).
Keywords:: Bernoulli polynomials, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.4.vii

24.4.34 $\frac{d}{dx}\mathop{B_{{n}}\/}\nolimits\!\left(x\right)=n\mathop{B_{{n-1}}\/}\nolimits\!\left(x\right),$ $n=1,2,\dots$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.5
Referenced by:: §24.13(i)
Permalink:: http://dlmf.nist.gov/24.4.E34
Encodings:: TeX, pMML, png

24.4.35 $\frac{d}{dx}\mathop{E_{{n}}\/}\nolimits\!\left(x\right)=n\mathop{E_{{n-1}}\/}\nolimits\!\left(x\right),$ $n=1,2,\dots$ .

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.5
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.4.E35
Encodings:: TeX, pMML, png

§24.4(viii) Symbolic Operations

Keywords:: Bernoulli polynomials, Euler polynomials, umbral calculus
Permalink:: http://dlmf.nist.gov/24.4.viii

Let $P(x)$ denote any polynomial in $x$ , and after expanding set $(B(x))^{n}=\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ and $(E(x))^{n}=\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ . Then