# periodic Bernoulli functions

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##### 2: 25.1 Special Notation
 $k,m,n$ nonnegative integers. … periodic Bernoulli function $B_{n}\left(x-\left\lfloor x\right\rfloor\right)$. …
##### 3: 25.11 Hurwitz Zeta Function
25.11.6 $\zeta\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-% \frac{s(s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{% (x+a)^{s+2}}\,\mathrm{d}x,$ $s\neq 1$, $\Re s>-1$, $a>0$.
25.11.7 $\zeta\left(s,a\right)=\frac{1}{a^{s}}+\frac{1}{(1+a)^{s}}\left(\frac{1}{2}+% \frac{1+a}{s-1}\right)+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}% \frac{B_{2k}}{2k}\frac{1}{(1+a)^{s+2k-1}}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}% \int_{1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{(x+a)^{s+2n+1}}\,% \mathrm{d}x,$ $s\neq 1$, $a>0$, $n=1,2,3,\dots$, $\Re s>-2n$.
25.11.19 $\zeta'\left(s,a\right)=-\frac{\ln a}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}% \right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(% \widetilde{B}_{2}\left(x\right)-B_{2})\ln\left(x+a\right)}{(x+a)^{s+2}}\,% \mathrm{d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x% \right)-B_{2}}{(x+a)^{s+2}}\,\mathrm{d}x,$ $\Re s>-1$, $s\neq 1$, $a>0$.
25.11.20 $(-1)^{k}{\zeta}^{(k)}\left(s,a\right)=\frac{(\ln a)^{k}}{a^{s}}\left(\frac{1}{% 2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_{r=0}^{k-1}\frac{(\ln a)^{r}}{r!(s-1)^{k% -r+1}}-\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)% -B_{2})(\ln\left(x+a\right))^{k}}{(x+a)^{s+2}}\,\mathrm{d}x+\frac{k(2s+1)}{2}% \int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a% \right))^{k-1}}{(x+a)^{s+2}}\,\mathrm{d}x-\frac{k(k-1)}{2}\int_{0}^{\infty}% \frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a\right))^{k-2}}{(x+a% )^{s+2}}\,\mathrm{d}x,$ $\Re s>-1$, $s\neq 1$, $a>0$.
##### 4: 24.17 Mathematical Applications
24.17.5 $M_{n}(x)=\begin{cases}\widetilde{B}_{n}\left(x\right)-B_{n},&n\text{ even},\\ \widetilde{B}_{n}\left(x+\frac{1}{2}\right),&n\text{ odd}.\end{cases}$
##### 5: 25.16 Mathematical Applications
25.16.6 $H\left(s\right)=-\zeta'\left(s\right)+\gamma\zeta\left(s\right)+\frac{1}{2}% \zeta\left(s+1\right)+\sum_{r=1}^{k}\zeta\left(1-2r\right)\zeta\left(s+2r% \right)+\sum_{n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{\infty}\frac{\widetilde{B}% _{2k+1}\left(x\right)}{x^{2k+2}}\,\mathrm{d}x,$
25.16.7 $H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-% 1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)% \zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+% 1}}\,\mathrm{d}x.$
##### 6: 25.2 Definition and Expansions
25.2.9 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}% {2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}% N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{% \widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,$ $\Re s>-2n$; $n,N=1,2,3,\dots$.
25.2.10 $\zeta\left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0% pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{% 1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,$ $\Re s>-2n$, $n=1,2,3,\dots$.
##### 7: 2.10 Sums and Sequences
As in §24.2, let $B_{n}$ and $B_{n}\left(x\right)$ denote the $n$th Bernoulli number and polynomial, respectively, and $\widetilde{B}_{n}\left(x\right)$ the $n$th Bernoulli periodic function $B_{n}\left(x-\left\lfloor x\right\rfloor\right)$. …
2.10.1 $\sum_{j=a}^{n}f(j)=\int_{a}^{n}f(x)\,\mathrm{d}x+\tfrac{1}{2}f(a)+\tfrac{1}{2}% f(n)+\sum_{s=1}^{m-1}\frac{B_{2s}}{(2s)!}\left(f^{(2s-1)}(n)-f^{(2s-1)}(a)% \right)+\int_{a}^{n}\frac{B_{2m}-\widetilde{B}_{2m}\left(x\right)}{(2m)!}f^{(2% m)}(x)\,\mathrm{d}x.$
2.10.5 $R_{m}(n)=\int_{n}^{\infty}\frac{\widetilde{B}_{2m}\left(x\right)-B_{2m}}{2m(2m% -1)x^{2m-1}}\,\mathrm{d}x.$
##### 8: Errata
• Equations (25.11.6), (25.11.19), and (25.11.20)

Originally all six integrands in these equations were incorrect because their numerators contained the function $\widetilde{B}_{2}\left(x\right)$. The correct function is $\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{2}$. The new equations are:

25.11.6 $\zeta\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-% \frac{s(s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{% (x+a)^{s+2}}\,\mathrm{d}x,$ $s\neq 1$, $\Re s>-1$, $a>0$

Reported 2016-05-08 by Clemens Heuberger.

25.11.19 $\zeta'\left(s,a\right)=-\frac{\ln a}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}% \right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(% \widetilde{B}_{2}\left(x\right)-B_{2})\ln\left(x+a\right)}{(x+a)^{s+2}}\,% \mathrm{d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x% \right)-B_{2}}{(x+a)^{s+2}}\,\mathrm{d}x,$ $\Re s>-1$, $s\neq 1$, $a>0$

Reported 2016-06-27 by Gergő Nemes.

25.11.20 $(-1)^{k}{\zeta}^{(k)}\left(s,a\right)=\frac{(\ln a)^{k}}{a^{s}}\left(\frac{1}{% 2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_{r=0}^{k-1}\frac{(\ln a)^{r}}{r!(s-1)^{k% -r+1}}-\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)% -B_{2})(\ln\left(x+a\right))^{k}}{(x+a)^{s+2}}\,\mathrm{d}x+\frac{k(2s+1)}{2}% \int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a% \right))^{k-1}}{(x+a)^{s+2}}\,\mathrm{d}x-\frac{k(k-1)}{2}\int_{0}^{\infty}% \frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a\right))^{k-2}}{(x+a% )^{s+2}}\,\mathrm{d}x,$ $\Re s>-1$, $s\neq 1$, $a>0$

Reported 2016-06-27 by Gergő Nemes.

• ##### 9: 24.16 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)).
##### 10: Bibliography B
• B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.