# series representation

(0.002 seconds)

## 1—10 of 74 matching pages

##### 1: 1.17 Integral and Series Representations of the Dirac Delta
###### §1.17(iii) SeriesRepresentations
By analogy with §1.17(ii) we have the formal series representation
##### 2: 25.16 Mathematical Applications
25.16.1 $\psi\left(x\right)=\sum_{m=1}^{\infty}\sum_{p^{m}\leq x}\ln p,$
25.16.5 $H\left(s\right)=\sum_{n=1}^{\infty}\frac{H_{n}}{n^{s}},$
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 $H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-% 1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)% \zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+% 1}}\mathrm{d}x.$
##### 4: 25.11 Hurwitz Zeta Function
###### §25.11(iv) SeriesRepresentations
25.11.8 $\zeta\left(s,\tfrac{1}{2}a\right)=\zeta\left(s,\tfrac{1}{2}a+\tfrac{1}{2}% \right)+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}},$ $\Re s>0$, $s\neq 1$, $0.
###### §25.11(x) Further SeriesRepresentations
25.11.37 $\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\zeta\left(nk,a\right)=-n\ln\Gamma\left(a% \right)+\ln\left(\prod_{j=0}^{n-1}\Gamma\left(a-e^{(2j+1)\pi i/n}\right)\right),$ $n=2,3,4,\dots$, $\Re a\geq 1$.
25.11.38 $\sum_{k=1}^{\infty}\genfrac{(}{)}{0.0pt}{}{n+k}{k}\zeta\left(n+k+1,a\right)z^{% k}=\frac{(-1)^{n}}{n!}\left({\psi}^{(n)}\left(a\right)-{\psi}^{(n)}\left(a-z% \right)\right),$ $n=1,2,3,\dots$, $\Re a>0$, $|z|<|a|$.
##### 5: 25.12 Polylogarithms
25.12.10 $\mathrm{Li}_{s}\left(z\right)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}.$
##### 6: 12.18 Methods of Computation
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …
##### 7: 25.15 Dirichlet $L$-functions
25.15.1 $L\left(s,\chi\right)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}},$ $\Re s>1$,
##### 8: 25.14 Lerch’s Transcendent
25.14.1 ${\Phi\left(z,s,a\right)\equiv\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$ $|z|<1$; $\Re s>1,|z|=1$.
##### 9: Bibliography V
• N. Ja. Vilenkin and A. U. Klimyk (1991) Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.
• N. Ja. Vilenkin and A. U. Klimyk (1992) Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.
• N. Ja. Vilenkin and A. U. Klimyk (1993) Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.
• H. Volkmer (2021) Fourier series representation of Ferrers function ${\sf P}$ .
• ##### 10: Bibliography E
• J. A. Ewell (1990) A new series representation for $\zeta(3)$ . Amer. Math. Monthly 97 (3), pp. 219–220.