# representations as sums of powers

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##### 2: 5.19 Mathematical Applications
As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. …
$S=\sum_{k=0}^{\infty}a_{k},$
Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. …
##### 3: 11.9 Lommel Functions
where $A$, $B$ are arbitrary constants, $s_{{\mu},{\nu}}\left(z\right)$ is the Lommel function defined by
11.9.3 $s_{{\mu},{\nu}}\left(z\right)=z^{\mu+1}\sum_{k=0}^{\infty}(-1)^{k}\frac{z^{2k}% }{a_{k+1}(\mu,\nu)},$
11.9.4 $a_{k}(\mu,\nu)=\prod_{m=1}^{k}\left((\mu+2m-1)^{2}-\nu^{2}\right)=4^{k}{\left(% \frac{\mu-\nu+1}{2}\right)_{k}}{\left(\frac{\mu+\nu+1}{2}\right)_{k}},$ $k=0,1,2,\dots$.
For collections of integral representations and integrals see Apelblat (1983, §12.17), Babister (1967, p. 85), Erdélyi et al. (1954a, §§4.19 and 5.17), Gradshteyn and Ryzhik (2000, §6.86), Marichev (1983, p. 193), Oberhettinger (1972, pp. 127–128, 168–169, and 188–189), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105–106 and 191–192), Oberhettinger and Badii (1973, §2.14), Prudnikov et al. (1990, §§1.6 and 2.9), Prudnikov et al. (1992a, §3.34), and Prudnikov et al. (1992b, §3.32).
##### 4: 27.13 Functions
Each representation of $n$ as a sum of elements of $S$ is called a partition of $n$, and the number $S(n)$ of such partitions is often of great interest. … This problem is named after Edward Waring who, in 1770, stated without proof and with limited numerical evidence, that every positive integer $n$ is the sum of four squares, of nine cubes, of nineteen fourth powers, and so on. … Hardy and Littlewood (1925) conjectures that $G\left(k\right)<2k+1$ when $k$ is not a power of 2, and that $G\left(k\right)\leq 4k$ when $k$ is a power of 2, but the most that is known (in 2009) is $G\left(k\right) for some constant $c$. …
###### §27.13(iv) Representation by Squares
Mordell (1917) notes that $r_{k}\left(n\right)$ is the coefficient of $x^{n}$ in the power-series expansion of the $k$th power of the series for $\vartheta\left(x\right)$. …
##### 5: 16.11 Asymptotic Expansions
$c_{k}=-\frac{1}{k\kappa^{\kappa}}\sum_{m=0}^{k-1}c_{m}e_{k,m},$ $k\geq 1$,
Explicit representations for the coefficients $c_{k}$ are given in Volkmer (2023). It may be observed that $H_{p,q}(z)$ represents the sum of the residues of the poles of the integrand in (16.5.1) at $s=-a_{j},-a_{j}-1,\dots$, $j=1,\dots,p$, provided that these poles are all simple, that is, no two of the $a_{j}$ differ by an integer. … Here the upper or lower signs are chosen according as $z$ lies in the upper or lower half-plane; in consequence, in the fractional powers4.2(iv)) of $ze^{\mp\pi i}$ its phases are $\operatorname{ph}z\mp\pi$, respectively. … Explicit representations for the coefficients $c_{k}$ are given in Volkmer and Wood (2014). …
##### 6: 7.17 Inverse Error Functions
###### §7.17(ii) Power Series
7.17.2 $\operatorname{inverf}x=t+\tfrac{1}{3}t^{3}+\tfrac{7}{30}t^{5}+\tfrac{127}{630}% t^{7}+\cdots=\sum_{m=0}^{\infty}a_{m}t^{2m+1},$ $|x|<1$,
7.17.2_5 $a_{m+1}=\frac{1}{2m+3}\sum_{n=0}^{m}\frac{2n+1}{m-n+1}a_{n}a_{m-n},$ $m=0,1,2,\ldots$.
For an alternative representation of (7.17.3) see Blair et al. (1976).
##### 7: 9.12 Scorer Functions
where …
###### §9.12(vii) Integral Representations
9.12.29 $\operatorname{Hi}\left(z\right)\sim-\frac{1}{\pi z}\sum_{k=0}^{\infty}\frac{(3% k)!}{k!(3z^{3})^{k}}+\frac{e^{\zeta}}{\sqrt{\pi}z^{1/4}}\sum_{k=0}^{\infty}% \frac{u_{k}}{\zeta^{k}},$ $|\operatorname{ph}z|\leq\pi-\delta$.
##### 8: 28.34 Methods of Computation
• (b)

Representations for $w_{\mbox{\tiny I}}(\pi;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$; see Schäfke (1961a).

• (a)

Summation of the power series in §§28.6(i) and 28.15(i) when $\left|q\right|$ is small.

• (a)

Summation of the power series in §§28.6(ii) and 28.15(ii) when $\left|q\right|$ is small.

• (d)

Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

• ##### 9: 27.4 Euler Products and Dirichlet Series
27.4.1 $\sum_{n=1}^{\infty}f(n)=\prod_{p}\left(1+\sum_{r=1}^{\infty}f(p^{r})\right),$
27.4.2 $\sum_{n=1}^{\infty}f(n)=\prod_{p}(1-f(p))^{-1}.$
The completely multiplicative function $f(n)=n^{-s}$ gives the Euler product representation of the Riemann zeta function $\zeta\left(s\right)$25.2(i)): …
27.4.11 $\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^{-s}=\zeta\left(s\right)% \zeta\left(s-\alpha\right),$ $\Re s>\max(1,1+\Re\alpha)$,
##### 10: 27.14 Unrestricted Partitions
A fundamental problem studies the number of ways $n$ can be written as a sum of positive integers $\leq n$, that is, the number of solutions of … Multiplying the power series for $\mathit{f}\left(x\right)$ with that for $1/\mathit{f}\left(x\right)$ and equating coefficients, we obtain the recursion formula … and $s(h,k)$ is a Dedekind sum given by … where $\varepsilon=\exp\left(\pi\mathrm{i}(((a+d)/(12c))-s(d,c))\right)$ and $s(d,c)$ is given by (27.14.11). … The 24th power of $\eta\left(\tau\right)$ in (27.14.12) with $e^{2\pi\mathrm{i}\tau}=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujan’s tau function $\tau\left(n\right)$: …