§24.14 Sums

Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.14

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

§24.14(i) Quadratic 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).
Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.14.i

24.14.1 $\sum _{{k=0}}^{n}{n\choose k}\mathop{B_{{k}}\/}\nolimits\!\left(x\right)\mathop{B_{{n-k}}\/}\nolimits\!\left(y\right)=n(x+y-1)\mathop{B_{{n-1}}\/}\nolimits\!\left(x+y\right)-(n-1)\mathop{B_{{n}}\/}\nolimits\!\left(x+y\right),$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : 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

24.14.2 $\sum _{{k=0}}^{n}{n\choose k}\mathop{B_{{k}}\/}\nolimits\mathop{B_{{n-k}}\/}\nolimits=(1-n)\mathop{B_{{n}}\/}\nolimits-n\mathop{B_{{n-1}}\/}\nolimits.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.14.E2
Encodings:: TeX, pMML, png

24.14.3 $\sum _{{k=0}}^{n}{n\choose k}\mathop{E_{{k}}\/}\nolimits\!\left(h\right)\mathop{E_{{n-k}}\/}\nolimits\!\left(x\right)=2(\mathop{E_{{n+1}}\/}\nolimits\!\left(x+h\right)-(x+h-1)\mathop{E_{{n}}\/}\nolimits\!\left(x+h\right)),$

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/24.14.E3
Encodings:: TeX, pMML, png

24.14.4 $\sum _{{k=0}}^{n}{n\choose k}\mathop{E_{{k}}\/}\nolimits\mathop{E_{{n-k}}\/}\nolimits=-2^{{n+1}}\mathop{E_{{n+1}}\/}\nolimits\!\left(0\right)=-2^{{n+2}}(1-2^{{n+2}})\frac{\mathop{B_{{n+2}}\/}\nolimits}{n+2}.$

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/24.14.E4
Encodings:: TeX, pMML, png

24.14.5 $\sum _{{k=0}}^{n}{n\choose k}\mathop{E_{{k}}\/}\nolimits\!\left(h\right)\mathop{B_{{n-k}}\/}\nolimits\!\left(x\right)=2^{n}\mathop{B_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}(x+h)\right),$

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/24.14.E5
Encodings:: TeX, pMML, png

24.14.6 $\sum _{{k=0}}^{n}{n\choose k}2^{k}\mathop{B_{{k}}\/}\nolimits\mathop{E_{{n-k}}\/}\nolimits=2(1-2^{{n-1}})\mathop{B_{{n}}\/}\nolimits-n\mathop{E_{{n-1}}\/}\nolimits.$

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
Referenced by:: §24.14(i)
Permalink:: http://dlmf.nist.gov/24.14.E6
Encodings:: TeX, pMML, png

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

24.14.7 $\sum _{{j=0}}^{m}\sum _{{k=0}}^{n}\binom{m}{j}\binom{n}{k}\frac{\mathop{B_{{j}}\/}\nolimits\mathop{B_{{k}}\/}\nolimits}{m+n-j-k+1}=(-1)^{{m-1}}\frac{m!n!}{(m+n)!}\mathop{B_{{m+n}}\/}\nolimits.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $!$ : $n!$ : factorial, $j$ : 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

§24.14(ii) 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).
Keywords:: Bernoulli numbers, Euler numbers, determinants
Permalink:: http://dlmf.nist.gov/24.14.ii

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 $\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!}\mathop{B_{{2j}}\/}\nolimits\mathop{B_{{2k}}\/}\nolimits\mathop{B_{{2\ell}}\/}\nolimits=(n-1)(2n-1)\mathop{B_{{2n}}\/}\nolimits+n(n-\tfrac{1}{2})\mathop{B_{{2n-2}}\/}\nolimits,$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $!$ : $n!$ : factorial, $j$ : 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

24.14.9 $\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!}\mathop{E_{{2j}}\/}\nolimits\mathop{E_{{2k}}\/}\nolimits\mathop{E_{{2\ell}}\/}\nolimits=\tfrac{1}{2}\left(\mathop{E_{{2n}}\/}\nolimits-\mathop{E_{{2n+2}}\/}\nolimits\right).$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $!$ : $n!$ : factorial, $j$ : 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

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 $\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!(2m)!}\mathop{B_{{2j}}\/}\nolimits\mathop{B_{{2k}}\/}\nolimits\mathop{B_{{2\ell}}\/}\nolimits\mathop{B_{{2m}}\/}\nolimits=-{2n+3\choose 3}\mathop{B_{{2n}}\/}\nolimits-\frac{4}{3}n^{2}(2n-1)\mathop{B_{{2n-2}}\/}\nolimits.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $!$ : $n!$ : factorial, $j$ : 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

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