# sums of squares

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## 11—20 of 42 matching pages

##### 11: 27.13 Functions
The basic problem is that of expressing a given positive integer $n$ as a sum of integers from some prescribed set $S$ whose members are primes, squares, cubes, or other special integers. … 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. … By similar methods Jacobi proved that $r_{4}\left(n\right)=8\sigma_{1}\left(n\right)$ if $n$ is odd, whereas, if $n$ is even, $r_{4}\left(n\right)=24$ times the sum of the odd divisors of $n$. …
##### 12: 26.12 Plane Partitions
where $\sigma_{2}(j)$ is the sum of the squares of the divisors of $j$. …
##### 13: 27.6 Divisor Sums
27.6.1 $\sum_{d\mathbin{|}n}\lambda\left(d\right)=\begin{cases}1,&n\mbox{ is a square}% ,\\ 0,&\mbox{otherwise}.\end{cases}$
##### 14: 26.18 Counting Techniques
26.18.1 $\left|S\setminus(A_{1}\cup A_{2}\cup\cdots\cup A_{n})\right|=\left|S\right|+% \sum_{t=1}^{n}(-1)^{t}\sum_{1\leq j_{1}
26.18.2 $N+\sum_{t=1}^{n}(-1)^{t}\sum_{1\leq j_{1}
With the notation of §26.15, the number of placements of $n$ nonattacking rooks on an $n\times n$ chessboard that avoid the squares in a specified subset $B$ is
26.18.4 $k^{n}+\sum_{t=1}^{n}(-1)^{t}\genfrac{(}{)}{0.0pt}{}{k}{t}(k-t)^{n}.$
##### 15: 28.30 Expansions in Series of Eigenfunctions
Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi]$ can be expanded in a uniformly and absolutely convergent series
28.30.3 $f(x)=\sum_{m=0}^{\infty}f_{m}w_{m}(x),$
##### 16: 22.11 Fourier and Hyperbolic Series
22.11.13 ${\operatorname{sn}}^{2}\left(z,k\right)=\frac{1}{k^{2}}\left(1-\frac{E}{K}% \right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{2n}}% \cos\left(2n\zeta\right).$
##### 17: Bibliography
• T. M. Apostol and H. S. Zuckerman (1951) On magic squares constructed by the uniform step method. Proc. Amer. Math. Soc. 2 (4), pp. 557–565.
• T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.
• T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.
• R. Askey and G. Gasper (1976) Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.
• R. Askey (1974) Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.
• ##### 18: 1.8 Fourier Series
1.8.1 $f(x)=\tfrac{1}{2}a_{0}+\sum^{\infty}_{n=1}(a_{n}\cos\left(nx\right)+b_{n}\sin% \left(nx\right)),$
1.8.3 $f(x)=\sum^{\infty}_{n=-\infty}c_{n}e^{inx},$
Then the series (1.8.1) converges to the sumwhen $f(x)$ and $g(x)$ are square-integrable and $a_{n},b_{n}$ and $a^{\prime}_{n},b^{\prime}_{n}$ are their respective Fourier coefficients. …
1.8.15 $\tfrac{1}{2}f(0)+\sum^{\infty}_{n=1}f(n)=\int^{\infty}_{0}f(x)\mathrm{d}x+2% \sum^{\infty}_{n=1}\int^{\infty}_{0}f(x)\cos\left(2\pi nx\right)\mathrm{d}x.$
3.5.5 $\int_{-\infty}^{\infty}f(t)\mathrm{d}t=h\sum_{k=-\infty}^{\infty}f(kh)+E_{h}(f),$
and the square $S$, given by $\left|x\right|\leq h$, $\left|y\right|\leq h$: …
14.28.1 $P_{\nu}\left(z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)% ^{1/2}\cos\phi\right)=P_{\nu}\left(z_{1}\right)P_{\nu}\left(z_{2}\right)+2\sum% _{m=1}^{\infty}(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1% \right)}\*P^{m}_{\nu}\left(z_{1}\right)P^{m}_{\nu}(z_{2})\cos\left(m\phi\right),$
where the branches of the square roots have their principal values when $z_{1},z_{2}\in(1,\infty)$ and are continuous when $z_{1},z_{2}\in\mathbb{C}\setminus(0,1]$. …
14.28.2 $\sum_{n=0}^{\infty}(2n+1)Q_{n}\left(z_{1}\right)P_{n}\left(z_{2}\right)=\frac{% 1}{z_{1}-z_{2}},$ $z_{1}\in\mathcal{E}_{1}$, $z_{2}\in\mathcal{E}_{2}$,