Articles of exponential sum

Sum of Gaussian Sequence

I am looking for a closed form or an estimation for the sum of a Gaussian sequence expressed as $$ \sum_{x=0}^{N-1} e^{\frac{-a}{N^2} \: x^2} $$ where $a$ is a constant positive integer. The interesting part is that I have simulated this sequence using MATLAB for $N=0$ to $10^6$ and found that the result is linear […]

Amount of real numbers, in a given sequence, whose fractional part lies in a given closed interval

Fix an interval $[a,b] \subset [0,1]$ and let $S$ be a given sequence of real numbers. Are there any ad-hoc methods that may be used to estimate $$ T := \#\{ s \in S \ : \ \{s\} \in [a,b]\}? $$ Here $\{s\} := s – \lfloor s \rfloor$ denotes the fractional part of $s$. […]

Summation evaluation of $\sum_{j=0}^k n^{1/2^j}$?

How do I go about solving this: $\sum_{j=0}^k n^{1/2^j}$ So, the terms of this series are $n , n^{1/2},n^{1/4},n^{1/8},…….n^{1/2^k}$ Any insights on what the thought process should be, to solve this? Any links to resources or even the name of this series would also help.. I initially thought the Power Series or the Puiseux Series […]

How to compute $\sum_{n=1}^N e^{-( n-c)^2}$

I have to compute or at least find good upper and lower bounds on \begin{align*} \sum_{n=1}^N e^{-( n-c)^2/b} \end{align*} and \begin{align*} \sum_{n=1}^N ne^{-( n-c)^2/b} \end{align*} where $c$ and $b\ge 0$ are constants. What I tried -I tried to rewrite the above two using hyperbolic function but this approach led to nothing. -this reminds of the […]

log of summation expression

I am curious about simplifying the following expression: $\log \left(\sum_\limits{i=0}^{n}x_i \right)$ Is there any rule to simplify a summation inside the log?

Expected time for the queue to become empty.

Question: Suppose, there is a queue A having i customers initially. The service time of the queue is Exponentially distributed with parameter $\mu$ (i.e., mean service time of one customer in Queue A is 1/$\mu$). Arrival to queue A has Poisson distribution with mean $\lambda$. What is the expected time before queue A becomes empty […]

Determining the Value of a Gauss Sum.

Can we evaluate the exact form of $$g\left(k,n\right)=\sum_{r=0}^{n-1}\exp\left(2\pi i\frac{r^{2}k}{n}\right) $$ for general $k$ and $n$? For $k=1$, on MathWorld we have that $$g\left(1,n\right)=\left\{ \begin{array}{cc} (1+i)\sqrt{n} & \ \text{when}\ n\equiv0\ \text{mod}\ 4\\ \sqrt{n} & \text{when}\ n\equiv1\ \text{mod}\ 4\\ 0 & \text{when}\ n\equiv2\ \text{mod}\ 4\\ i\sqrt{n} & \text{when}\ n\equiv3\ \text{mod}\ 4 \end{array}\right\} .$$ I know how to […]

Plotting exponential partial sums in the complex plane

I was plotting the following sequence of points $(a_n)_{n = 0}^\infty$ in the complex plane for various reals $\alpha > 1$: $$a_n = \sum_{k = 0}^n e^{i k^\alpha}$$ I found that for many values of $\alpha > 1$ the path traced by $(a_n)$ displayed interesting nonrandom and even sometimes chaotic behaviour. The following images show […]

Closed form for an infinite sum over Gamma functions?

I am having quite a bit of trouble trying to find a closed form (or a really fast way to compute) for the infinite sum $$\sum_{n=1}^{\infty} a^n \dfrac{\gamma(n+1,b)}{\Gamma(n+1)\Gamma(n)}$$ where $\gamma(n,b) = \int_0^bz^{n-1}e^{-z} dz$ is the lower incomplete Gamma function and $a,b > 0$. Any ideas how to solve this?

An alternating series identity with a hidden hyperbolic tangent

How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi – 5\log(2))$$ The identity follows from MO189199 and the penultimate identity on this page (for $x=\frac{1}{2}$). Scratch-work: I computed the Mellin transform of $$f(x) = \frac{(-1)^x}{x(e^{x\pi} + 1)}$$ and re-wrote the function in terms of […]