# infinite series expansions

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##### 1: 24.8 Series Expansions
###### §24.8(i) Fourier Series
If $n=1,2,\dots$ and $0\leq x\leq 1$, then …
###### §24.8(ii) Other Series
24.8.9 $E_{2n}=(-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{\cosh\left(\tfrac{1}{2}\pi k% \right)}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^{2n}}{e^{2\pi(2k+1)}-1},$ $n=1,2,\dots$.
##### 2: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
###### §22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
22.12.13 $2K\operatorname{cs}\left(2Kt,k\right)=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}% \frac{\pi}{\tan\left(\pi(t-n\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)% ^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t-m-n\tau}\right).$
##### 3: 25.2 Definition and Expansions
###### §25.2(ii) Other InfiniteSeries
25.2.5 $\gamma_{n}=\lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln k)^{n}}{k}-\frac{(% \ln m)^{n+1}}{n+1}\right).$
##### 4: Bibliography S
• H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
• ##### 5: 31.11 Expansions in Series of Hypergeometric Functions
###### §31.11(v) Doubly-InfiniteSeries
Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions. …
##### 6: 25.12 Polylogarithms
The cosine series in (25.12.7) has the elementary sum … For real or complex $s$ and $z$ the polylogarithm $\mathrm{Li}_{s}\left(z\right)$ is defined by … For each fixed complex $s$ the series defines an analytic function of $z$ for $|z|<1$. The series also converges when $|z|=1$, provided that $\Re s>1$. … The notation $\phi\left(z,s\right)$ was used for $\mathrm{Li}_{s}\left(z\right)$ in Truesdell (1945) for a series treated in Jonquière (1889), hence the alternative name Jonquière’s function. …
##### 7: 16.11 Asymptotic Expansions
16.11.7 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q}F_{q}}% \left({a_{1},\dots,a_{q}\atop b_{1},\dots,b_{q}};z\right)\sim H_{q,q}(z{% \mathrm{e}}^{\mp\pi\mathrm{i}})+E_{q,q}(z).$
16.11.8 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q-1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q-1}F_{q}}% \left({a_{1},\dots,a_{q-1}\atop b_{1},\dots,b_{q}};-z\right)\sim H_{q-1,q}(z)+% E_{q-1,q}(ze^{-\pi\mathrm{i}})+E_{q-1,q}(ze^{\pi\mathrm{i}}),$
16.11.9 $\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{p}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{p}F_{q}}% \left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};-z\right)\sim E_{p,q}(ze^{-% \pi\mathrm{i}})+E_{p,q}(ze^{\pi\mathrm{i}}),$
##### 8: 18.38 Mathematical Applications
In consequence, expansions of functions that are infinitely differentiable on $[-1,1]$ in series of Chebyshev polynomials usually converge extremely rapidly. …
##### 9: 2.1 Definitions and Elementary Properties
In those cases it is usually necessary to interpret each infinite series separately in the manner described above; that is, it is not always possible to reinterpret the asymptotic approximation as a single asymptotic expansion. …
##### 10: 28.11 Expansions in Series of Mathieu Functions
###### §28.11 Expansions in Series of Mathieu Functions
Let $f(z)$ be a $2\pi$-periodic function that is analytic in an open doubly-infinite strip $S$ that contains the real axis, and $q$ be a normal value (§28.7). …The series (28.11.1) converges absolutely and uniformly on any compact subset of the strip $S$. See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of $q$ see Meixner et al. (1980, p. 33). …
28.11.7 $\sin(2m+2)z=\sum_{n=0}^{\infty}B_{2m+2}^{2n+2}(q)\mathrm{se}_{2n+2}\left(z,q% \right).$