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Let $f(x) = e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}-1 =\sum_{i=1}^N e^{k_i/x}-1 $, and let $K = \sum_{i=1}^N k_i $. The restrictions that $x > 0$ and $k_i < 0$ are important in what follows. $f'(x) =\sum -\frac{k_i}{x^2}e^{k_i/x} =-\frac1{x^2}\sum k_ie^{k_i/x} =\frac1{x^2}\sum |k_i|e^{k_i/x} $, so $f'(x) > 0$. This means that your function has at ...


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Write an algorithm to solve the problem... Let $$ Q = \left( \sum_{m=1}^{n} \exp(k_m y) - 1 \right)^2. $$ In case we have the solution, we have $$ Q = 0 $$. For changes in $Q$ we have $$ \delta Q = 2 Q \left( \sum_{m=1}^{n} k_m \exp(k_m y) \right) \delta y. $$ For each step use $$ \delta y = - 2 Q \left( \sum_{m=1}^{n} k_m \exp(k_m y) ...


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This is equivalent to finding roots of the polynomial $$y^{-k_1}+\cdots +y^{-k_N}-1=0$$ (by making $y:= e^{-\frac{1}{x}}$) so your problem is as difficult as finding roots of polynomials. It depends on the degree of that polynomial, in this case, it depends on the max of the $-k_i$.


3

This is related to Jacobi Theta functions since $$\sum_{n=-\infty}^\infty e^{-n^2\pi x}=\vartheta _3\left(0,e^{-\pi x}\right)$$ $$\sum_{n=-\infty}^\infty e^{-n^2\pi / x}=\vartheta _3\left(0,e^{-\frac{\pi }{x}}\right)$$ and the function $$\Psi(x)=\frac 12 \Big(\vartheta _3\left(0,e^{-\pi x}\right)-1\Big)$$ satisfies $$\frac{1+2\Psi(x)}{1+2\Psi(\frac ...


2

The sequence is closely related to what is called a Zadoff-Chu sequence, or Frank-Zadoff-Chu (FZC) sequence or just a Chu sequence depending on who is doing the name-calling. One of the properties of FZC sequences is that their Discrete Fourier Transforms are another FZC sequence, conjugated, scaled, and possibly time-scaled as well. These sequences are ...


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You could probably use the generalized quadratic Gauss sum reciprocity formula, defined for integers $a, b, c$ satisfying $ac \neq 0$ and $ac+b$ even. In this case, we set $\displaystyle S(a,b,c) = \sum_{x=0}^{|c|-1} e\left(\frac{ax^2+bx}{2c}\right)$, and the theorem states \begin{eqnarray*} S(a,b,c) = \left|\frac{c}a\right| ...



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