24 Bernoulli and Euler Polynomials Properties 24.15 Related Sequences of Numbers 24.17 Mathematical Applications

§24.16 Generalizations

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Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials, generalizations, generalized
Permalink:: http://dlmf.nist.gov/24.16
See also:: Annotations for Ch.24

§24.16(i) Higher-Order Analogs

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Notes:: For (24.16.6), (24.16.7), and (24.16.8) see Howard (1996a).
Referenced by:: §24.11, §5.11(iii), §8.18(ii), Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/24.16.i
Addition (effective with 1.0.11):: References to López and Temme (1999b, 2010b) were added into the second sentence below (24.16.4).
See also:: Annotations for §24.16 and Ch.24

Polynomials and Numbers of Integer Order

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Defines:: $B^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Bernoulli polynomials and $E^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Euler polynomials
See also:: Annotations for §24.16(i), §24.16 and Ch.24

For $\ell=0,1,2,\dotsc$ , Bernoulli and Euler polynomials of order $\ell$ are defined respectively by

24.16.1	$\displaystyle\left(\frac{t}{e^{t}-1}\right)^{\ell}e^{xt}$	$\displaystyle=\sum_{n=0}^{\infty}B^{(\ell)}_{n}\left(x\right)\frac{t^{n}}{n!},$
$\|t\|<2\pi$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $B^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Bernoulli polynomials, $\ell$ : integer, $n$ : integer, $t$ : real or complex and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.16.E1 Encodings: TeX, pMML, png See also: Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24
24.16.2	$\displaystyle\left(\frac{2}{e^{t}+1}\right)^{\ell}e^{xt}$	$\displaystyle=\sum_{n=0}^{\infty}E^{(\ell)}_{n}\left(x\right)\frac{t^{n}}{n!},$
$\|t\|<\pi$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $E^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Euler polynomials, $\ell$ : integer, $n$ : integer, $t$ : real or complex and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.16.E2 Encodings: TeX, pMML, png See also: Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

When $x=0$ they reduce to the Bernoulli and Euler numbers of order $\ell$ :

24.16.3	$\displaystyle B^{(\ell)}_{n}$	$\displaystyle=B^{(\ell)}_{n}\left(0\right),$
24.16.3	$\displaystyle E^{(\ell)}_{n}$	$\displaystyle=E^{(\ell)}_{n}\left(0\right).$
ⓘ Symbols: $B^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Bernoulli polynomials, $E^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Euler polynomials, $\ell$ : integer and $n$ : integer Referenced by: §24.16(i) Permalink: http://dlmf.nist.gov/24.16.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

Also for $\ell=1,2,3,\dotsc$ ,

24.16.4		$\left(\frac{\ln\left(1+t\right)}{t}\right)^{\ell}=\ell\sum_{n=0}^{\infty}\frac% {B^{(\ell+n)}_{n}}{\ell+n}\frac{t^{n}}{n!},$
$\|t\|<1$ .
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $B^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Bernoulli polynomials, $\ln\NVar{z}$ : principal branch of logarithm function, $\ell$ : integer, $n$ : integer and $t$ : real or complex Referenced by: §24.16(i) Permalink: http://dlmf.nist.gov/24.16.E4 Encodings: TeX, pMML, png See also: Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

For this and other properties see Milne-Thomson (1933, pp. 126–153) or Nörlund (1924, pp. 144–162).

For extensions of $B^{(\ell)}_{n}\left(x\right)$ to complex values of $x$ , $n$ , and $\ell$ , and also for uniform asymptotic expansions for large $x$ and large $n$ , see Temme (1995b) and López and Temme (1999b, 2010b).

Bernoulli Numbers of the Second Kind

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Keywords:: Bernoulli numbers, of the second kind
See also:: Annotations for §24.16(i), §24.16 and Ch.24

24.16.5		$\frac{t}{\ln\left(1+t\right)}=\sum_{n=0}^{\infty}b_{n}t^{n},$
$\|t\|<1$ ,
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : integer, $t$ : real or complex and $b_{n}$ Permalink: http://dlmf.nist.gov/24.16.E5 Encodings: TeX, pMML, png See also: Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

24.16.6		$n!b_{n}=-\frac{1}{n-1}B^{(n-1)}_{n},$
$n=2,3,\dots$ .
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $B^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Bernoulli polynomials, $n$ : integer and $b_{n}$ Referenced by: §24.16(i) Permalink: http://dlmf.nist.gov/24.16.E6 Encodings: TeX, pMML, png See also: Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

Degenerate Bernoulli Numbers

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Keywords:: Bernoulli numbers, degenerate
See also:: Annotations for §24.16(i), §24.16 and Ch.24

For sufficiently small $|t|$ ,

24.16.7		$\frac{t}{(1+\lambda t)^{\ifrac{1}{\lambda}}-1}=\sum_{n=0}^{\infty}\beta_{n}(% \lambda)\frac{t^{n}}{n!},$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $n$ : integer, $t$ : real or complex and $\beta_{n}(\lambda)$ Referenced by: §24.16(i) Permalink: http://dlmf.nist.gov/24.16.E7 Encodings: TeX, pMML, png See also: Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

24.16.8		$\beta_{n}(\lambda)=n!b_{n}\lambda^{n}+\sum_{k=1}^{\left\lfloor\ifrac{n}{2}% \right\rfloor}\frac{n}{2k}B_{2k}s\left(n-1,2k-1\right)\lambda^{n-2k},$
$n=2,3,\dots$ .
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $k$ : integer, $n$ : integer and $\beta_{n}(\lambda)$ Referenced by: §24.16(i) Permalink: http://dlmf.nist.gov/24.16.E8 Encodings: TeX, pMML, png See also: Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

Here $s\left(n,m\right)$ again denotes the Stirling number of the first kind.

Nörlund Polynomials

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Keywords:: Nörlund polynomials
See also:: Annotations for §24.16(i), §24.16 and Ch.24

24.16.9		$\left(\frac{t}{e^{t}-1}\right)^{x}=\sum_{n=0}^{\infty}B^{(x)}_{n}\frac{t^{n}}{% n!},$
$\|t\|<2\pi$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $B^{(\NVar{\ell})}_{\NVar{n}}\left(\NVar{x}\right)$ : generalized Bernoulli polynomials, $n$ : integer, $t$ : real or complex and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.16.E9 Encodings: TeX, pMML, png See also: Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

$B^{(x)}_{n}$ is a polynomial in $x$ of degree $n$ . (This notation is consistent with (24.16.3) when $x=\ell$ .)

§24.16(ii) Character Analogs

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Keywords:: conductor, generalized Bernoulli polynomials, primitive Dirichlet characters, relation to generalized Bernoulli polynomials
Notes:: See Washington (1997, pp. 31–34). For (24.16.12) and (24.16.13) see Dilcher (1988, pp. 8 and 9).
Permalink:: http://dlmf.nist.gov/24.16.ii
See also:: Annotations for §24.16 and Ch.24

Let $\chi$ be a primitive Dirichlet character $\mod f$ (see §27.8). Then $f$ is called the conductor of $\chi$ . Generalized Bernoulli numbers and polynomials belonging to $\chi$ are defined by

24.16.10		$\sum_{a=1}^{f}\frac{\chi(a)te^{at}}{e^{ft}-1}=\sum_{n=0}^{\infty}B_{n,\chi}% \frac{t^{n}}{n!},$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : integer, $t$ : real or complex, $\chi(a)$ : Dirichlet character, $f$ : conductor and $B_{n,\chi}$ : Generalized Bernoulli numbers Permalink: http://dlmf.nist.gov/24.16.E10 Encodings: TeX, pMML, png See also: Annotations for §24.16(ii), §24.16 and Ch.24

24.16.11		$B_{n,\chi}(x)=\sum_{k=0}^{n}{n\choose k}B_{k,\chi}x^{n-k}.$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer, $x$ : real or complex, $\chi(a)$ : Dirichlet character and $B_{n,\chi}$ : Generalized Bernoulli numbers Permalink: http://dlmf.nist.gov/24.16.E11 Encodings: TeX, pMML, png See also: Annotations for §24.16(ii), §24.16 and Ch.24

Let $\chi_{0}$ be the trivial character and $\chi_{4}$ the unique (nontrivial) character with $f=4$ ; that is, $\chi_{4}(1)=1$ , $\chi_{4}(3)=-1$ , $\chi_{4}(2)=\chi_{4}(4)=0$ . Then

24.16.12		$B_{n}\left(x\right)=B_{n,\chi_{0}}(x-1),$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $n$ : integer, $x$ : real or complex, $\chi(a)$ : Dirichlet character and $B_{n,\chi}$ : Generalized Bernoulli numbers Referenced by: §24.16(ii) Permalink: http://dlmf.nist.gov/24.16.E12 Encodings: TeX, pMML, png See also: Annotations for §24.16(ii), §24.16 and Ch.24

24.16.13		$E_{n}\left(x\right)=-\frac{2^{1-n}}{n+1}B_{n+1,\chi_{4}}(2x-1).$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $n$ : integer, $x$ : real or complex, $\chi(a)$ : Dirichlet character and $B_{n,\chi}$ : Generalized Bernoulli numbers Referenced by: §24.16(ii) Permalink: http://dlmf.nist.gov/24.16.E13 Encodings: TeX, pMML, png See also: Annotations for §24.16(ii), §24.16 and Ch.24

For further properties see Berndt (1975a).

§24.16(iii) Other Generalizations

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Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials, generalizations, generalized
Permalink:: http://dlmf.nist.gov/24.16.iii
See also:: Annotations for §24.16 and Ch.24

In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); $p$ -adic integer order Bernoulli numbers (Adelberg (1996)); $p$ -adic $q$ -Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).

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