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 $\sum_{k=0}^{n}{n\choose k}B_{k}\left(x\right)B_{n-k}\left(y\right)=n(x+y-1)B_{% n-1}\left(x+y\right)-(n-1)B_{n}\left(x+y\right),$

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Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : integer, : integer and : 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 $\sum_{k=0}^{n}{n\choose k}B_{k}B_{n-k}=(1-n)B_{n}-nB_{n-1}.$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : integer and : 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 $\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)E_{n-k}\left(x\right)=2(E_{n+1}% \left(x+h\right)-(x+h-1)E_{n}\left(x+h\right)),$

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Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : integer, : integer and : 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 $\sum_{k=0}^{n}{n\choose k}E_{k}E_{n-k}=-2^{n+1}E_{n+1}\left(0\right)=-2^{n+2}(% 1-2^{n+2})\frac{B_{n+2}}{n+2}.$

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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, : integer and : 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 $\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)B_{n-k}\left(x\right)=2^{n}B_{n}% \left(\tfrac{1}{2}(x+h)\right),$

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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, : integer, : integer and : 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 $\sum_{k=0}^{n}{n\choose k}2^{k}B_{k}E_{n-k}=2(1-2^{n-1})B_{n}-nE_{n-1}.$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : integer and : 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 be even with and nonzero. Then

24.14.7 $\sum_{j=0}^{m}\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{}{m}{j}\genfrac{(}{)}{0.0pt}% {}{n}{k}\frac{B_{j}B_{k}}{m+n-j-k+1}=(-1)^{m-1}\frac{m!n!}{(m+n)!}B_{m+n}.$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : factorial (as in ), : integer, : integer, : integer and : 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 $\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!}B_{2j}B_{2k}B_{2\ell}=(n-1)(2n-1)B_{2n}+n(% n-\tfrac{1}{2})B_{2n-2},$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, : factorial (as in ), : integer, : integer, $\ell$ : integer and : 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 $\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!}E_{2j}E_{2k}E_{2\ell}=\tfrac{1}{2}\left(E_% {2n}-E_{2n+2}\right).$

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, : factorial (as in ), : integer, : integer, $\ell$ : integer and : 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 $\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!(2m)!}B_{2j}B_{2k}B_{2\ell}B_{2m}=-{2n+3% \choose 3}B_{2n}-\frac{4}{3}n^{2}(2n-1)B_{2n-2}.$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : factorial (as in ), : integer, : integer, $\ell$ : integer, : integer and : 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 and column for $r,s=0,1,\dots,n$ . Then