24 Bernoulli and Euler Polynomials Properties 24.14 Sums 24.16 Generalizations

§24.15 Related Sequences of Numbers

Permalink:: http://dlmf.nist.gov/24.15

§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

Notes:: See Dumont and Viennot (1980).
Keywords:: Bernoulli numbers, Genocchi numbers
Permalink:: http://dlmf.nist.gov/24.15.i

24.15.1 $\frac{2t}{e^{t}+1}=\sum_{{n=1}}^{\infty}G_{n}\frac{t^{n}}{n!},$

Symbols:: : base of exponential function, : : factorial, : integer, : real or complex and $G_{n}$ : Genocchi numbers
Permalink:: http://dlmf.nist.gov/24.15.E1
Encodings:: TeX, pMML, png

24.15.2 $G_{n}=2(1-2^{n})\mathop{B_{{n}}\/}\nolimits.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, : integer and $G_{n}$ : Genocchi numbers
Permalink:: http://dlmf.nist.gov/24.15.E2
Encodings:: TeX, pMML, png

See Table 24.15.1.

§24.15(ii) Tangent Numbers

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

24.15.3 $\mathop{\tan\/}\nolimits t=\sum_{{n=0}}^{\infty}T_{n}\frac{t^{n}}{n!},$

Symbols:: : : factorial, $\mathop{\tan\/}\nolimits z$ : tangent function, : integer, : 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

24.15.4 $T_{{2n-1}}=(-1)^{{n-1}}\frac{2^{{2n}}(2^{{2n}}-1)}{2n}\mathop{B_{{2n}}\/}\nolimits,$ $n=1,2,\dots$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, : integer and $T_{n}$ : tangent numbers
Referenced by:: §24.19(i)
Permalink:: http://dlmf.nist.gov/24.15.E4
Encodings:: TeX, pMML, png

24.15.5 $T_{{2n}}=0,$ $n=0,1,\dots$ .

Symbols:: : integer and $T_{n}$ : tangent numbers
Permalink:: http://dlmf.nist.gov/24.15.E5
Encodings:: TeX, pMML, png

Table 24.15.1: Genocchi and Tangent numbers.

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

Symbols:: : integer, $G_{n}$ : Genocchi numbers and $T_{n}$ : tangent numbers
Keywords:: Genocchi numbers, tangent numbers
Referenced by:: §24.15(i)
Permalink:: http://dlmf.nist.gov/24.15.T1

§24.15(iii) Stirling 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).
Keywords:: Bernoulli numbers, Stirling numbers (first and second kinds)
Referenced by:: §26.8(vi)
Permalink:: http://dlmf.nist.gov/24.15.iii

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

24.15.6 $\mathop{B_{{n}}\/}\nolimits=\sum_{{k=0}}^{n}(-1)^{k}\frac{k!\mathop{S\/}% \nolimits\!\left(n,k\right)}{k+1},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, : : factorial, : integer and : integer
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E6
Encodings:: TeX, pMML, png

24.15.7 $\mathop{B_{{n}}\/}\nolimits=\sum_{{k=0}}^{n}(-1)^{k}\binomial{n+1}{k+1}\mathop% {S\/}\nolimits\!\left(n+k,k\right)\bigg/\binomial{n+k}{k},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, $\binom{m}{n}$ : binomial coefficient, : integer and : integer
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E7
Encodings:: TeX, pMML, png

24.15.8 $\sum_{{k=0}}^{n}(-1)^{{n+k}}\mathop{s\/}\nolimits\!\left(n+1,k+1\right)\mathop% {B_{{k}}\/}\nolimits=\frac{n!}{n+1}.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind, : : factorial, : integer and : integer
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E8
Encodings:: TeX, pMML, png

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

24.15.9 $p\frac{\mathop{B_{{n}}\/}\nolimits}{n}\equiv\mathop{S\/}\nolimits\!\left(p-1+n% ,p-1\right)\;\;(\mathop{{\rm mod}}p^{2}),$ $1\leq n\leq p-2$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, : integer and : prime
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E9
Encodings:: TeX, pMML, png

24.15.10 $\frac{2n-1}{4n}p^{2}\mathop{B_{{2n}}\/}\nolimits\equiv{\mathop{S\/}\nolimits\!% \left(p+2n,p-1\right)\;\;(\mathop{{\rm mod}}p^{3})},$ $2\leq 2n\leq p-3$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind, : integer and : prime
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E10
Encodings:: TeX, pMML, png

§24.15(iv) Fibonacci and Lucas Numbers

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

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 $\sum_{{k=0}}^{{\left\lfloor\ifrac{n}{2}\right\rfloor}}{n\choose 2k}\left(\frac% {5}{9}\right)^{k}\mathop{B_{{2k}}\/}\nolimits u_{{n-2k}}=\frac{n}{6}v_{{n-1}}+% \frac{n}{3^{n}}v_{{2n-2}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $\left\lfloor x\right\rfloor$ : floor of , : integer, : integer, $u_{n}$ : Fibonacci numbers and $v_{n}$ : Lucas numbers
Permalink:: http://dlmf.nist.gov/24.15.E11
Encodings:: TeX, pMML, png

24.15.12 $\sum_{{k=0}}^{{\left\lfloor\ifrac{n}{2}\right\rfloor}}{n\choose 2k}\left(\frac% {5}{4}\right)^{k}\mathop{E_{{2k}}\/}\nolimits v_{{n-2k}}=\frac{1}{2^{{n-1}}}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : binomial coefficient, $\left\lfloor x\right\rfloor$ : floor of , : integer, : integer and $v_{n}$ : Lucas numbers
Permalink:: http://dlmf.nist.gov/24.15.E12
Encodings:: TeX, pMML, png

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