§24.5 Recurrence Relations

24.5.1		$$\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k\left(x\right)=n{x}^{n-1},$$
$n=2,3,\mathrm{\dots }$,
ⓘ Symbols: $B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex Permalink: http://dlmf.nist.gov/24.5.E1 Encodings: TeX, pMML, png See also: Annotations for §24.5(i), §24.5 and Ch.24

24.5.2		$$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(x\right)+E_n\left(x\right)=2x^n,$$
$n=1,2,\mathrm{\dots }$.
ⓘ Symbols: $E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex Permalink: http://dlmf.nist.gov/24.5.E2 Encodings: TeX, pMML, png See also: Annotations for §24.5(i), §24.5 and Ch.24

24.5.3		$$\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k=0,$$
$n=2,3,\mathrm{\dots }$,
ⓘ Symbols: $B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer Referenced by: §24.19(i) Permalink: http://dlmf.nist.gov/24.5.E3 Encodings: TeX, pMML, png See also: Annotations for §24.5(i), §24.5 and Ch.24

24.5.4		$$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{2n}{2k}\right)E_{2k}=0,$$
$n=1,2,\mathrm{\dots }$,
ⓘ Symbols: $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer Referenced by: §24.19(i) Permalink: http://dlmf.nist.gov/24.5.E4 Encodings: TeX, pMML, png See also: Annotations for §24.5(i), §24.5 and Ch.24

24.5.5		$$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)2^k{E}_{n-k}+E_n=2.$$
ⓘ Symbols: $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer Permalink: http://dlmf.nist.gov/24.5.E5 Encodings: TeX, pMML, png See also: Annotations for §24.5(i), §24.5 and Ch.24

24.5.6		$$\sum _{k=2}^n\left(\genfrac{}{}{0pt}{}{n}{k-2}\right)\frac{B_k}{k}=\frac{1}{(n+1)(n+2)}-B_{n+1},$$
$n=2,3,\mathrm{\dots }$,
ⓘ Symbols: $B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer Referenced by: §24.5(ii) Permalink: http://dlmf.nist.gov/24.5.E6 Encodings: TeX, pMML, png See also: Annotations for §24.5(ii), §24.5 and Ch.24

24.5.7		$$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{B_k}{n+2-k}=\frac{B_{n+1}}{n+1},$$
$n=1,2,\mathrm{\dots }$,
ⓘ Symbols: $B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer Referenced by: §24.5(ii) Permalink: http://dlmf.nist.gov/24.5.E7 Encodings: TeX, pMML, png See also: Annotations for §24.5(ii), §24.5 and Ch.24

24.5.8		$$\sum _{k=0}^n\frac{2^{2k}{B}_{2k}}{(2k)!(2n+1-2k)!}=\frac{1}{(2n)!},$$
$n=1,2,\mathrm{\dots }$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $!$: factorial (as in $n!$), $k$: integer and $n$: integer Referenced by: §24.5(ii) Permalink: http://dlmf.nist.gov/24.5.E8 Encodings: TeX, pMML, png See also: Annotations for §24.5(ii), §24.5 and Ch.24

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

24.5.9		$a_n$	$={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\displaystyle \frac{b_{n-k}}{k+1}},$
		$b_n$	$={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)B_k{a}_{n-k}.$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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 See also: Annotations for §24.5(iii), §24.5 and Ch.24
24.5.10		$a_n$	$={\displaystyle \sum _{k=0}^{\lfloor n/2\rfloor }}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2k}}\right)b_{n-2k},$
		$b_n$	$={\displaystyle \sum _{k=0}^{\lfloor n/2\rfloor }}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2k}}\right)E_{2k}{a}_{n-2k}.$
ⓘ Symbols: $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: 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 See also: Annotations for §24.5(iii), §24.5 and Ch.24