# periodic Euler functions

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##### 2: 24.17 Mathematical Applications
24.17.2 $R_{m}(n)=\frac{1}{2(m-1)!}\int_{a}^{n}f^{(m)}(x)\widetilde{E}_{m-1}\left(h-x% \right)\,\mathrm{d}x.$
24.17.3 $S_{n}(x)=\frac{\widetilde{E}_{n}\left(x+\tfrac{1}{2}n+\tfrac{1}{2}\right)}{% \widetilde{E}_{n}\left(\tfrac{1}{2}n+\tfrac{1}{2}\right)},$ $n=0,1,\dots$,
##### 3: 25.13 Periodic Zeta Function
25.13.2 $F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(2\pi)^{1-s}}\*\left(e^{\pi i(% 1-s)/2}\zeta\left(1-s,x\right)+e^{\pi i(s-1)/2}\zeta\left(1-s,1-x\right)\right),$ $0, $\Re s>1$,
25.13.3 $\zeta\left(1-s,x\right)=\frac{\Gamma\left(s\right)}{(2\pi)^{s}}\left(e^{-\pi is% /2}F\left(x,s\right)+e^{\pi is/2}F\left(-x,s\right)\right),$ $\Re s>0$ if $0; $\Re s>1$ if $x=1$.
##### 4: 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 $H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-% 1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)% \zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+% 1}}\,\mathrm{d}x.$
##### 5: 25.1 Special Notation
(For other notation see Notation for the Special Functions.)
 $k,m,n$ nonnegative integers. … periodic Bernoulli function $B_{n}\left(x-\left\lfloor x\right\rfloor\right)$. …
The main function treated in this chapter is the Riemann zeta function $\zeta\left(s\right)$. … The main related functions are the Hurwitz zeta function $\zeta\left(s,a\right)$, the dilogarithm $\operatorname{Li}_{2}\left(z\right)$, the polylogarithm $\operatorname{Li}_{s}\left(z\right)$ (also known as Jonquière’s function $\phi\left(z,s\right)$), Lerch’s transcendent $\Phi\left(z,s,a\right)$, and the Dirichlet $L$-functions $L\left(s,\chi\right)$.
##### 6: 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)).
##### 7: 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 $\zeta\left(s,a\right)=\frac{1}{a^{s}}+\frac{1}{(1+a)^{s}}\left(\frac{1}{2}+% \frac{1+a}{s-1}\right)+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}% \frac{B_{2k}}{2k}\frac{1}{(1+a)^{s+2k-1}}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}% \int_{1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{(x+a)^{s+2n+1}}\,% \mathrm{d}x,$ $s\neq 1$, $a>0$, $n=1,2,3,\dots$, $\Re s>-2n$.
##### 8: 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 $\zeta\left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0% pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{% 1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,$ $\Re s>-2n$, $n=1,2,3,\dots$.
##### 9: 32.10 Special Function Solutions
###### §32.10 Special Function Solutions
If $\gamma\delta\neq 0$, then as in §32.2(ii) we may set $\gamma=1$ and $\delta=-1$. … where the fundamental periods $2\phi_{1}$ and $2\phi_{2}$ are linearly independent functions satisfying the hypergeometric equation …The solution (32.10.34) is an essentially transcendental function of both constants of integration since $\mbox{P}_{\mbox{\scriptsize VI}}$ with $\alpha=\beta=\gamma=0$ and $\delta=\tfrac{1}{2}$ does not admit an algebraic first integral of the form $P(z,w,w^{\prime},C)=0$, with $C$ a constant. …
##### 10: 2.10 Sums and Sequences
###### §2.10(i) Euler–Maclaurin Formula
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 $B_{n}\left(x-\left\lfloor x\right\rfloor\right)$. … From §24.12(i), (24.2.2), and (24.4.27), $\widetilde{B}_{2m}\left(x\right)-B_{2m}$ is of constant sign $(-1)^{m}$. … where $\gamma$ is Euler’s constant (§5.2(ii)) and $\zeta'$ is the derivative of the Riemann zeta function25.2(i)). … For extensions of the Euler–Maclaurin formula to functions $f(x)$ with singularities at $x=a$ or $x=n$ (or both) see Sidi (2004, 2012b, 2012a). …