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
§24.15(ii) Tangent Numbers
§24.15(iii) Stirling Numbers
§24.15(iv) Fibonacci and Lucas Numbers

§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	$\displaystyle\sum_{k=0}^{n}(-1)^{n+k}s\left(n+1,k+1\right)B_{k}$	$\displaystyle=\frac{n!}{n+1}.$
ⓘ 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.

§24.15 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