24 Bernoulli and Euler Polynomials Properties 24.14 Sums 24.16 Generalizations

§24.15 Related Sequences of Numbers

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Permalink:: http://dlmf.nist.gov/24.15
See also:: Annotations for Ch.24

§24.15(i) Genocchi Numbers

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Keywords:: Bernoulli numbers, Genocchi numbers, relations to
Notes:: See Dumont and Viennot (1980).
Permalink:: http://dlmf.nist.gov/24.15.i
See also:: Annotations for §24.15 and Ch.24

24.15.1	$\displaystyle\frac{2t}{e^{t}+1}$	$\displaystyle=\sum_{n=1}^{\infty}G_{n}\frac{t^{n}}{n!},$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : integer, $t$ : real or complex and $G_{n}$ : Genocchi numbers Permalink: http://dlmf.nist.gov/24.15.E1 Encodings: TeX, pMML, png See also: Annotations for §24.15(i), §24.15 and Ch.24
24.15.2	$\displaystyle G_{n}$	$\displaystyle=2(1-2^{n})B_{n}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $n$ : integer and $G_{n}$ : Genocchi numbers Permalink: http://dlmf.nist.gov/24.15.E2 Encodings: TeX, pMML, png See also: Annotations for §24.15(i), §24.15 and Ch.24

See Table 24.15.1.

§24.15(ii) Tangent Numbers

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Keywords:: Bernoulli numbers, relations to, tangent numbers
Notes:: See Graham et al. (1994, Ch. 6) and Knuth and Buckholtz (1967).
Permalink:: http://dlmf.nist.gov/24.15.ii
See also:: Annotations for §24.15 and Ch.24

24.15.3		$\tan t=\sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!},$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\tan\NVar{z}$ : tangent function, $n$ : integer, $t$ : real or complex and $T_{n}$ : tangent numbers Referenced by: §24.19(i) Permalink: http://dlmf.nist.gov/24.15.E3 Encodings: TeX, pMML, png See also: Annotations for §24.15(ii), §24.15 and Ch.24

24.15.4		$T_{2n-1}=(-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}B_{2n},$
$n=1,2,\dots$ ,
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $n$ : integer and $T_{n}$ : tangent numbers Referenced by: §24.19(i) Permalink: http://dlmf.nist.gov/24.15.E4 Encodings: TeX, pMML, png See also: Annotations for §24.15(ii), §24.15 and Ch.24

24.15.5		$T_{2n}=0,$
$n=0,1,\dots$ .
ⓘ Symbols: $n$ : integer and $T_{n}$ : tangent numbers Permalink: http://dlmf.nist.gov/24.15.E5 Encodings: TeX, pMML, png See also: Annotations for §24.15(ii), §24.15 and Ch.24

Table 24.15.1: Genocchi and Tangent numbers.

$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$G_{n}$	$0$	$1$	$-1$	$0$	$1$	$0$	$-3$	$0$	$17$
$T_{n}$	$0$	$1$	$0$	$2$	$0$	$16$	$0$	$272$	$0$

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Symbols:: $n$ : integer, $G_{n}$ : Genocchi numbers and $T_{n}$ : tangent numbers
Keywords:: Genocchi numbers, table, tables, tangent numbers
Referenced by:: §24.15(i)
Permalink:: http://dlmf.nist.gov/24.15.T1
See also:: Annotations for §24.15(ii), §24.15 and Ch.24

§24.15(iii) Stirling Numbers

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Keywords:: Bernoulli numbers, Stirling numbers, Stirling numbers (first and second kinds), relations to, relations to Bernoulli numbers
Notes:: For (24.15.6) and (24.15.8) see Todorov (1984, pp. 310 and 343). For (24.15.6) and (24.15.7) see Gould (1972, pp. 44 and 48).
Referenced by:: §26.8(vi), Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/24.15.iii
See also:: Annotations for §24.15 and Ch.24

The Stirling numbers of the first kind $s\left(n,m\right)$ , and the second kind $S\left(n,m\right)$ , are as defined in §26.8(i).

24.15.6	$\displaystyle B_{n}$	$\displaystyle=\sum_{k=0}^{n}(-1)^{k}\frac{k!S\left(n,k\right)}{k+1},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $!$ : factorial (as in $n!$ ), $k$ : integer and $n$ : integer Referenced by: §24.15(iii) Permalink: http://dlmf.nist.gov/24.15.E6 Encodings: TeX, pMML, png See also: Annotations for §24.15(iii), §24.15 and Ch.24
24.15.7	$\displaystyle B_{n}$	$\displaystyle=\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n+1}{k+1}S\left(n+% k,k\right)\bigg{/}\genfrac{(}{)}{0.0pt}{}{n+k}{k},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer Referenced by: §24.15(iii) Permalink: http://dlmf.nist.gov/24.15.E7 Encodings: TeX, pMML, png See also: Annotations for §24.15(iii), §24.15 and Ch.24
24.15.8	$\displaystyle\sum_{k=0}^{n}(-1)^{n+k}s\left(n+1,k+1\right)B_{k}=\frac{n!}{n+1}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $!$ : factorial (as in $n!$ ), $k$ : integer and $n$ : integer Referenced by: §24.15(iii) Permalink: http://dlmf.nist.gov/24.15.E8 Encodings: TeX, pMML, png See also: Annotations for §24.15(iii), §24.15 and Ch.24

In (24.15.9) and (24.15.10) $p$ denotes a prime. See Horata (1991).

24.15.9		$p\frac{B_{n}}{n}\equiv S\left(p-1+n,p-1\right)\pmod{p^{2}},$
$1\leq n\leq p-2$ ,
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\equiv$ : modular equivalence, $n$ : integer and $p$ : prime Referenced by: §24.15(iii) Permalink: http://dlmf.nist.gov/24.15.E9 Encodings: TeX, pMML, png See also: Annotations for §24.15(iii), §24.15 and Ch.24

24.15.10		$\frac{2n-1}{4n}p^{2}B_{2n}\equiv{S\left(p+2n,p-1\right)\pmod{p^{3}}},$
$2\leq 2n\leq p-3$ .
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\equiv$ : modular equivalence, $n$ : integer and $p$ : prime Referenced by: §24.15(iii) Permalink: http://dlmf.nist.gov/24.15.E10 Encodings: TeX, pMML, png See also: Annotations for §24.15(iii), §24.15 and Ch.24

§24.15(iv) Fibonacci and Lucas Numbers

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Keywords:: Fibonacci numbers, Lucas numbers
Notes:: See Kelisky (1957, pp. 32 and 34).
Permalink:: http://dlmf.nist.gov/24.15.iv
See also:: Annotations for §24.15 and Ch.24

The Fibonacci numbers are defined by $u_{0}=0$ , $u_{1}=1$ , and $u_{n+1}=u_{n}+u_{n-1}$ , $n\geq 1$ . The Lucas numbers are defined by $v_{0}=2$ , $v_{1}=1$ , and $v_{n+1}=v_{n}+v_{n-1}$ , $n\geq 1$ .

24.15.11	$\displaystyle\sum_{k=0}^{\left\lfloor\ifrac{n}{2}\right\rfloor}{n\choose 2k}% \left(\frac{5}{9}\right)^{k}B_{2k}u_{n-2k}$	$\displaystyle=\frac{n}{6}v_{n-1}+\frac{n}{3^{n}}v_{2n-2},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $k$ : integer, $n$ : integer, $u_{n}$ : Fibonacci numbers and $v_{n}$ : Lucas numbers Permalink: http://dlmf.nist.gov/24.15.E11 Encodings: TeX, pMML, png See also: Annotations for §24.15(iv), §24.15 and Ch.24
24.15.12	$\displaystyle\sum_{k=0}^{\left\lfloor\ifrac{n}{2}\right\rfloor}{n\choose 2k}% \left(\frac{5}{4}\right)^{k}E_{2k}v_{n-2k}$	$\displaystyle=\frac{1}{2^{n-1}}.$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $k$ : integer, $n$ : integer and $v_{n}$ : Lucas numbers Permalink: http://dlmf.nist.gov/24.15.E12 Encodings: TeX, pMML, png See also: Annotations for §24.15(iv), §24.15 and Ch.24

For further information on the Fibonacci numbers see §26.11.

24.14 Sums 24.16 Generalizations

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§24.15 Related Sequences of Numbers

Contents

§24.15(i) Genocchi Numbers

§24.15(ii) Tangent Numbers

§24.15(iii) Stirling Numbers

§24.15(iv) Fibonacci and Lucas Numbers

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