§24.5 Recurrence Relations

Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.5

§24.5(i) Basic Relations

Notes:: See Erdélyi et al. (1953a, pp. 35–38) or Nörlund (1924, pp. 19 and 24).
Permalink:: http://dlmf.nist.gov/24.5.i

24.5.1 $\sum _{{k=0}}^{{n-1}}{n\choose k}\mathop{B_{{k}}\/}\nolimits\!\left(x\right)=nx^{{n-1}},$ $n=2,3,\dots$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.5.E1
Encodings:: TeX, pMML, png

24.5.2 $\sum _{{k=0}}^{n}{n\choose k}\mathop{E_{{k}}\/}\nolimits\!\left(x\right)+\mathop{E_{{n}}\/}\nolimits\!\left(x\right)=2x^{n},$ $n=1,2,\dots$ .

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.5.E2
Encodings:: TeX, pMML, png

24.5.3 $\sum _{{k=0}}^{{n-1}}{n\choose k}\mathop{B_{{k}}\/}\nolimits=0,$ $n=2,3,\dots$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer and $n$ : integer
Referenced by:: §24.19(i)
Permalink:: http://dlmf.nist.gov/24.5.E3
Encodings:: TeX, pMML, png

24.5.4 $\sum _{{k=0}}^{n}{2n\choose 2k}\mathop{E_{{2k}}\/}\nolimits=0,$ $n=1,2,\dots$ ,

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer and $n$ : integer
Referenced by:: §24.19(i)
Permalink:: http://dlmf.nist.gov/24.5.E4
Encodings:: TeX, pMML, png

24.5.5 $\sum _{{k=0}}^{n}{n\choose k}2^{k}\mathop{E_{{n-k}}\/}\nolimits+\mathop{E_{{n}}\/}\nolimits=2.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.5.E5
Encodings:: TeX, pMML, png

Notes:: For (24.5.6) see Apostol (2008). For (24.5.7) and (24.5.8) see Apostol (1976, p. 275).
Keywords:: Bernoulli numbers
Permalink:: http://dlmf.nist.gov/24.5.ii

24.5.6 $\sum _{{k=2}}^{n}{n\choose k-2}\frac{\mathop{B_{{k}}\/}\nolimits}{k}=\frac{1}{(n+1)(n+2)}-\mathop{B_{{n+1}}\/}\nolimits,$ $n=2,3,\dots$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer and $n$ : integer
Referenced by:: §24.5(ii)
Permalink:: http://dlmf.nist.gov/24.5.E6
Encodings:: TeX, pMML, png

24.5.7 $\sum _{{k=0}}^{n}{n\choose k}\frac{\mathop{B_{{k}}\/}\nolimits}{n+2-k}=\frac{\mathop{B_{{n+1}}\/}\nolimits}{n+1},$ $n=1,2,\dots$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer and $n$ : integer
Referenced by:: §24.5(ii)
Permalink:: http://dlmf.nist.gov/24.5.E7
Encodings:: TeX, pMML, png

24.5.8 $\sum _{{k=0}}^{n}\frac{2^{{2k}}\mathop{B_{{2k}}\/}\nolimits}{(2k)!(2n+1-2k)!}=\frac{1}{(2n)!},$ $n=1,2,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $!$ : $n!$ : factorial, $k$ : integer and $n$ : integer
Referenced by:: §24.5(ii)
Permalink:: http://dlmf.nist.gov/24.5.E8
Encodings:: TeX, pMML, png

In each of (24.5.9) and (24.5.10) the first identity implies the second one and vice-versa.

24.5.9

$a_{n}=\sum _{{k=0}}^{n}{n\choose k}\frac{b_{{n-k}}}{k+1},$

$b_{n}=\sum _{{k=0}}^{n}{n\choose k}\mathop{B_{{k}}\/}\nolimits a_{{n-k}}.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer and $n$ : integer
Referenced by:: §24.5(iii)
Permalink:: http://dlmf.nist.gov/24.5.E9
Encodings:: TeX, TeX, pMML, pMML, png, png

24.5.10

$a_{n}=\sum _{{k=0}}^{{\left\lfloor\ifrac{n}{2}\right\rfloor}}{n\choose 2k}b_{{n-2k}},$

$b_{n}=\sum _{{k=0}}^{{\left\lfloor\ifrac{n}{2}\right\rfloor}}{n\choose 2k}\mathop{E_{{2k}}\/}\nolimits a_{{n-2k}}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : binomial coefficient, $\left\lfloor x\right\rfloor$ : floor of $x$ , $k$ : integer and $n$ : integer
Referenced by:: §24.5(iii)
Permalink:: http://dlmf.nist.gov/24.5.E10
Encodings:: TeX, TeX, pMML, pMML, png, png