24 Bernoulli and Euler Polynomials Properties 24.13 Integrals 24.15 Related Sequences of Numbers

§24.14 Sums

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Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials, sums
Permalink:: http://dlmf.nist.gov/24.14
See also:: Annotations for Ch.24

§24.14(i) Quadratic Recurrence Relations
§24.14(ii) Higher-Order Recurrence Relations
§24.14(iii) Compendia

§24.14(i) Quadratic Recurrence Relations

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Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials, quadratic, quadratic and higher order, recurrence relations
Notes:: For (24.14.1)–(24.14.6) see Nörlund (1922, pp. 135–142). For (24.14.7) see Carlitz (1961a, p. 992).
Permalink:: http://dlmf.nist.gov/24.14.i
See also:: Annotations for §24.14 and Ch.24

24.14.1	$\displaystyle\sum_{k=0}^{n}{n\choose k}B_{k}\left(x\right)B_{n-k}\left(y\right)$	$\displaystyle=n(x+y-1)B_{n-1}\left(x+y\right)-(n-1)B_{n}\left(x+y\right),$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex Referenced by: §24.14(i) Permalink: http://dlmf.nist.gov/24.14.E1 Encodings: TeX, pMML, png See also: Annotations for §24.14(i), §24.14 and Ch.24
24.14.2	$\displaystyle\sum_{k=0}^{n}{n\choose k}B_{k}B_{n-k}$	$\displaystyle=(1-n)B_{n}-nB_{n-1}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer Permalink: http://dlmf.nist.gov/24.14.E2 Encodings: TeX, pMML, png See also: Annotations for §24.14(i), §24.14 and Ch.24
24.14.3	$\displaystyle\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)E_{n-k}\left(x\right)$	$\displaystyle=2(E_{n+1}\left(x+h\right)-(x+h-1)E_{n}\left(x+h\right)),$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.14.E3 Encodings: TeX, pMML, png See also: Annotations for §24.14(i), §24.14 and Ch.24
24.14.4	$\displaystyle\sum_{k=0}^{n}{n\choose k}E_{k}E_{n-k}$	$\displaystyle=-2^{n+1}E_{n+1}\left(0\right)=-2^{n+2}(1-2^{n+2})\frac{B_{n+2}}{% n+2}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer Permalink: http://dlmf.nist.gov/24.14.E4 Encodings: TeX, pMML, png See also: Annotations for §24.14(i), §24.14 and Ch.24
24.14.5	$\displaystyle\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)B_{n-k}\left(x\right)$	$\displaystyle=2^{n}B_{n}\left(\tfrac{1}{2}(x+h)\right),$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.14.E5 Encodings: TeX, pMML, png See also: Annotations for §24.14(i), §24.14 and Ch.24
24.14.6	$\displaystyle\sum_{k=0}^{n}{n\choose k}2^{k}B_{k}E_{n-k}$	$\displaystyle=2(1-2^{n-1})B_{n}-nE_{n-1}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer Referenced by: §24.14(i) Permalink: http://dlmf.nist.gov/24.14.E6 Encodings: TeX, pMML, png See also: Annotations for §24.14(i), §24.14 and Ch.24

Let $m+n$ be even with $m$ and $n$ nonzero. Then

24.14.7		$\sum_{j=0}^{m}\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{}{m}{j}\genfrac{(}{)}{0.0pt}% {}{n}{k}\frac{B_{j}B_{k}}{m+n-j-k+1}=(-1)^{m-1}\frac{m!n!}{(m+n)!}B_{m+n}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $j$ : integer, $k$ : integer, $m$ : integer and $n$ : integer Referenced by: §24.14(i) Permalink: http://dlmf.nist.gov/24.14.E7 Encodings: TeX, pMML, png See also: Annotations for §24.14(i), §24.14 and Ch.24

§24.14(ii) Higher-Order Recurrence Relations

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Keywords:: Bernoulli numbers, Euler numbers, Hankel, determinants, persymmetric, quadratic and higher order, recurrence relations
Notes:: For (24.14.8) and (24.14.10) see Dilcher (1996, p. 27). (24.14.9) is a special case of Dilcher (1996, p. 36). See also Sitaramachandrarao and Davis (1986) and Huang and Huang (1999).
Permalink:: http://dlmf.nist.gov/24.14.ii
See also:: Annotations for §24.14 and Ch.24

In the following two identities, valid for $n\geq 2$ , the sums are taken over all nonnegative integers $j,k,\ell$ with $j+k+\ell=n$ .

24.14.8	$\displaystyle\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!}B_{2j}B_{2k}B_{2\ell}$	$\displaystyle=(n-1)(2n-1)B_{2n}+n(n-\tfrac{1}{2})B_{2n-2},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $!$ : factorial (as in $n!$ ), $j$ : integer, $k$ : integer, $\ell$ : integer and $n$ : integer Referenced by: §24.14(ii) Permalink: http://dlmf.nist.gov/24.14.E8 Encodings: TeX, pMML, png See also: Annotations for §24.14(ii), §24.14 and Ch.24
24.14.9	$\displaystyle\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!}E_{2j}E_{2k}E_{2\ell}$	$\displaystyle=\tfrac{1}{2}\left(E_{2n}-E_{2n+2}\right).$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $!$ : factorial (as in $n!$ ), $j$ : integer, $k$ : integer, $\ell$ : integer and $n$ : integer Referenced by: §24.14(ii) Permalink: http://dlmf.nist.gov/24.14.E9 Encodings: TeX, pMML, png See also: Annotations for §24.14(ii), §24.14 and Ch.24

In the next identity, valid for $n\geq 4$ , the sum is taken over all positive integers $j,k,\ell,m$ with $j+k+\ell+m=n$ .

24.14.10		$\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!(2m)!}B_{2j}B_{2k}B_{2\ell}B_{2m}=-{2n+3% \choose 3}B_{2n}-\frac{4}{3}n^{2}(2n-1)B_{2n-2}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $j$ : integer, $k$ : integer, $\ell$ : integer, $m$ : integer and $n$ : integer Referenced by: §24.14(ii) Permalink: http://dlmf.nist.gov/24.14.E10 Encodings: TeX, pMML, png See also: Annotations for §24.14(ii), §24.14 and Ch.24

For (24.14.11) and (24.14.12), see Al-Salam and Carlitz (1959). These identities can be regarded as higher-order recurrences. Let $\det[a_{r+s}]$ denote a Hankel (or persymmetric) determinant, that is, an $(n+1)\times(n+1)$ determinant with element $a_{r+s}$ in row $r$ and column $s$ for $r,s=0,1,\dots,n$ . Then

24.14.11	$\displaystyle\det[B_{r+s}]$	$\displaystyle=(-1)^{n(n+1)/2}\left(\prod_{k=1}^{n}k!\right)^{6}\Bigg{/}\left(% \prod_{k=1}^{2n+1}k!\right),$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\det$ : determinant, $!$ : factorial (as in $n!$ ), $k$ : integer, $n$ : integer, $r$ : row and $s$ : column Referenced by: §24.14(ii) Permalink: http://dlmf.nist.gov/24.14.E11 Encodings: TeX, pMML, png See also: Annotations for §24.14(ii), §24.14 and Ch.24
24.14.12	$\displaystyle\det[E_{r+s}]$	$\displaystyle=(-1)^{n(n+1)/2}\left(\prod_{k=1}^{n}k!\right)^{2}.$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $\det$ : determinant, $!$ : factorial (as in $n!$ ), $k$ : integer, $n$ : integer, $r$ : row and $s$ : column Referenced by: §24.14(ii) Permalink: http://dlmf.nist.gov/24.14.E12 Encodings: TeX, pMML, png See also: Annotations for §24.14(ii), §24.14 and Ch.24

§24.14(iii) Compendia

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Permalink:: http://dlmf.nist.gov/24.14.iii
See also:: Annotations for §24.14 and Ch.24

For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).

§24.14 Sums

Contents

§24.14(i) Quadratic Recurrence Relations

§24.14(ii) Higher-Order Recurrence Relations

§24.14(iii) Compendia