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

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

 24.16.3 $\displaystyle\mathop{B^{(\ell)}_{n}\/}\nolimits$ $\displaystyle=\mathop{B^{(\ell)}_{n}\/}\nolimits\!\left(0\right),$ $\displaystyle\mathop{E^{(\ell)}_{n}\/}\nolimits$ $\displaystyle=\mathop{E^{(\ell)}_{n}\/}\nolimits\!\left(0\right).$

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

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

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

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

# Bernoulli Numbers of the Second Kind

 24.16.5 $\frac{t}{\mathop{\ln\/}\nolimits\!\left(1+t\right)}=\sum_{n=0}^{\infty}b_{n}t^% {n},$ $|t|<1$,
 24.16.6 $n!b_{n}=-\frac{1}{n-1}\mathop{B^{(n-1)}_{n}\/}\nolimits,$ $n=2,3,\dots$.

# Degenerate Bernoulli Numbers

For sufficiently small $|t|$,

 24.16.7 $\frac{t}{(1+\lambda t)^{\ifrac{1}{\lambda}}-1}=\sum_{n=0}^{\infty}\beta_{n}(% \lambda)\frac{t^{n}}{n!},$
 24.16.8 $\beta_{n}(\lambda)=n!b_{n}\lambda^{n}+\sum_{k=1}^{\left\lfloor\ifrac{n}{2}% \right\rfloor}\frac{n}{2k}\mathop{B_{2k}\/}\nolimits\mathop{s\/}\nolimits\!% \left(n-1,2k-1\right)\lambda^{n-2k},$ $n=2,3,\dots$.

Here $\mathop{s\/}\nolimits\!\left(n,m\right)$ again denotes the Stirling number of the first kind.

# Nörlund Polynomials

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

$\mathop{B^{(x)}_{n}\/}\nolimits$ 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 $\mathop{B_{n}\/}\nolimits\!\left(x\right)=B_{n,\chi_{0}}(x-1),$
 24.16.13 $\mathop{E_{n}\/}\nolimits\!\left(x\right)=-\frac{2^{1-n}}{n+1}B_{n+1,\chi_{4}}% (2x-1).$

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 (1954b), 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)).