# §24.16 Generalizations

## §24.16(i) Higher-Order Analogs

### Polynomials and Numbers of Integer Order

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

 24.16.1 $\displaystyle\left(\frac{t}{e^{t}-1}\right)^{\ell}e^{xt}$ $\displaystyle=\sum_{n=0}^{\infty}B^{(\ell)}_{n}\left(x\right)\frac{t^{n}}{n!},$ $|t|<2\pi$, 24.16.2 $\displaystyle\left(\frac{2}{e^{t}+1}\right)^{\ell}e^{xt}$ $\displaystyle=\sum_{n=0}^{\infty}E^{(\ell)}_{n}\left(x\right)\frac{t^{n}}{n!},$ $|t|<\pi$.

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

 24.16.3 $\displaystyle B^{(\ell)}_{n}$ $\displaystyle=B^{(\ell)}_{n}\left(0\right),$ $\displaystyle E^{(\ell)}_{n}$ $\displaystyle=E^{(\ell)}_{n}\left(0\right).$ ⓘ 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 24

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

 24.16.4 $\left(\frac{\ln\left(1+t\right)}{t}\right)^{\ell}=\ell\sum_{n=0}^{\infty}\frac% {B^{(\ell+n)}_{n}}{\ell+n}\frac{t^{n}}{n!},$ $|t|<1$.

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

 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 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 24

### Degenerate Bernoulli Numbers

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 24
 24.16.8 $\beta_{n}(\lambda)=n!b_{n}\lambda^{n}+\sum_{k=1}^{\left\lfloor\ifrac{n}{2}% \right\rfloor}\frac{n}{2k}B_{2k}s\left(n-1,2k-1\right)\lambda^{n-2k},$ $n=2,3,\dots$.

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

### Nörlund Polynomials

 24.16.9 $\left(\frac{t}{e^{t}-1}\right)^{x}=\sum_{n=0}^{\infty}B^{(x)}_{n}\frac{t^{n}}{% n!},$ $|t|<2\pi$.

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

## §24.16(ii) Character Analogs

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 $\sum_{a=1}^{f}\frac{\chi(a)te^{at}}{e^{ft}-1}=\sum_{n=0}^{\infty}B_{n,\chi}% \frac{t^{n}}{n!},$
 24.16.11 $B_{n,\chi}(x)=\sum_{k=0}^{n}{n\choose k}B_{k,\chi}x^{n-k}.$

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 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 24

For further properties see Berndt (1975a).

## §24.16(iii) Other Generalizations

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)).