The closed-form-expressions tag has no wiki summary.

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### Closed form for convolution of two-dimensional Gaussian with characteristic function of a disk

Is there a closed form expression for the convolution of a two-dimensional (elliptical) Gaussian function with the characteristic function of the interior of an ellipse?
The motivation is that I have ...

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**1**answer

104 views

### Is it possible to get an equation with two exponentials and a bessel function in closed form?

Is it possible to get the equation below into closed form? I have tried using integration tables but I haven't found anything that matches. Are there any other methods to achieve a closed form ...

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72 views

### The hypertriangular function of $n$

I'm looking for papers or recent results on the hypertriangular function of $n$:
$$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$
This is A001923 in the OEIS.
I don't have much experience with ...

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315 views

### Does this equation has a closed-form solution for $t$? ($(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i)$)

We are given $n\in \mathbb N^+$ and $p\in[\frac{1}{2},\frac{n+1}{n+2}]$.
Our goal is to find $t\in[0,1]$ such that
$$(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i$$
Is there a closed-form ...

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284 views

### What summations of elementary trig functions are known to have (elementary) closed forms?

I've been trying to find a closed form of $\displaystyle \sum_k{\tan{(k)}}$ that contains only elementary functions, and I think I may be onto something. But rather than reinvent the wheel, I want to ...

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49 views

### Closed-for expression for Newton-Girard symmetric polynomials with 0/1 variables

There are $n$ Bernoulli $s_i\in\left\{0,1\right\}$, $i=1,...,n$ with equal marginals $\Pr(s_i=1)=\theta$ $\forall i$ so that E$(s_i)=\theta$. Their standardized mean deviations are
\begin{equation*}
...

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### Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?

I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here:
Is it possible to express ...

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**1**answer

347 views

### Conjectured closed form for definite integral

Let $K(x)$ be the complete elliptic integral of the first kind
(the argument is the parameter $m = k^2$).
Let $$ A = \int_0^1 \arcsin(K(x)) dx$$
With precision $1000$ decimal digits $\Re A = ...

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389 views

### Sum of series $a^{i^2}$

Is there any closed form known for the expression $\sum_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!

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577 views

### Random walk by simplex vertices

I apologize if this question is well-known, but I was unable to find it mentioned anywhere.
There exists a bug which moves around in $r$-space. The bug begins at the origin of this $r$-space. If the ...

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228 views

### Permutations of letters under some conditions

Let $W(p,q,r,s)$ be the number of permutations of the letters which satisfy the following conditions :
Condition 1 : The letters are consist of $P,Q,R,S$.
Condition 2 : The number of letter ...

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**1**answer

160 views

### Infinite series - analytical solution

Analytical Solution is required for:
$$\sum_{n=0}^\infty (2n+1)\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty (2n+1)^2\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty n(n+1)(2n+1)\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty ...

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**1**answer

163 views

### Generic way to solve f(x+1) - f(x) = g(x) when g(x) is given [closed]

All
I have been looking around for a general way to solve the problem of $f(x+1) - f(x) = g(x)$, where $g(x)$ is given. Has this problem been studied before?
If there does not exist such a general ...

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**1**answer

382 views

### sum of integral part of n/k

Is there any direct formula or algorithm better than the brute force (O(n) algorithm by iterating from 1 to n) way to calculate the sum
\begin{equation}
S = \sum\limits_{i=1}^n [{\frac{n}{i}}]
...

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177 views

### looking for f(x) in f(x) = a.exp((x-f(x))/b) [closed]

not a math expert, but this problem is really bugging me. whichever way i turn it, i cant find an expression for f(x) which satisfies
f(x) = exp(( x - f(x)) /b ).
I can also express the problem as ...

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**1**answer

324 views

### Convergence and Closed Form of an Integral Involving Bell Numbers

1. Does the following integral converge ?
$$\int_0^\infty \frac{b(x)}{B(x)} dx$$
where
$$b(x) = \sum_{n=1}^\infty \frac{n^x}{n^n} \qquad and \qquad B(x) = \sum_{n=1}^\infty \frac{n^x}{n!}$$
2. ...

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123 views

### multivariate integral calculation in closed form

I am looking for a closed form for the below integral but since I don't have the necessary backgrounds I am not able to solve it: i know the final solution is in the form of modified Bessel functions ...

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131 views

### Integrate the exponential of sum of circular differences?

Given positive integer $N$ and parameters $T>0$, $a$, $b$, what is
$\int_{t_1=0}^T \cdots \int_{t_N=0}^T e^{a(t_1+\cdots+t_N)+b(|t_1-t_2|+\cdots+|t_{i-1}-t_i|+|t_N-t_1|)} dt_1 \cdots dt_N$ ?
Any ...

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**1**answer

220 views

### An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$

Consider the following series:
$$S=\sum_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$
It can be expressed in terms of a hypergeometric function:
...

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105 views

### Estimate the scale of the power series with Poisson pdf/pmf-like terms

I would like to have an estimate for the series
$$P(t) = \sum\limits_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^m,$$
where $e$ is the base of natural logarithm, $k!$ is the factorial of the integer $k$, ...

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269 views

### Any closed form for series like $F(x)=\Sigma_{i=p}^{\infty}x^p$,p is prime$?

Any closed form for series like $$F(x)=\Sigma_{i=2}^{\infty}x^p,\text{p is prime}$$ or $$F(x)=\Sigma_{i=0}^{\infty}x^{i!}$$?
More generally,we can obtain a power series from decimal expansion of a ...

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### How closed-form conjectures are made?

Recently I posted a conjecture at Math.SE:
$$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$
where $J_\mu(x)$ and $Y_\mu(x)$ ...

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647 views

### Is this combination of generalized polygamma and dilogarithm actually zero? $\Im\;\psi^{(-2)}(1+i)+\frac1{4\pi}\text{Li}_2(e^{-2\pi})-\log\sqrt{2\pi}+\frac{5\pi}{24}+\frac12$

I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^{-4\times10^3}$ in ...

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97 views

### Fitting algebraic expression to a number [algorithm]

I know that it may turn out useless, but this is precisely the reason why I'm asking.
Does any one know of an existing piece of code that would find me the best approximation to a given irrational ...

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458 views

### Is there a closed form expression/series expansion for $\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$ ?

I've been trying to find a closed form expression/series expansion for the following integral without success:
$$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...

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776 views

### $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

I'm trying to solve the integral
$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$,
where $s$, $r$ and $m$>1 are positive integers.
My question is whether a closed form ...

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**1**answer

330 views

### Experimental mathematics: how are floating point equations discovered/converted to exact equations?

the 2005 AMS article/survey on experimental mathematics[1] by Bailey/Borwein mentions many remarkable successes in the field including new formulas for $\pi$ that were discovered via the PSLQ ...

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180 views

### Expectation under a t-distribution

Given parameters $\lambda, \nu>0$, a covariance matrix $R$, a mean vector $\mu \in R^p$, the Arellano-Valle and Bolfarine's generalized $t$ distribution is given by (see, for example, the book by ...

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192 views

### Growth of a particular sequence

Not sure if this is research level? While analyzing a particular algorithm, I came across the following series:
$\sum_{k=1}^{m-1}\frac{3^{k}}{\prod_{i=1}^{k-1}2^{2^{i}}}$
Is there a closed form ...

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258 views

### an equation in fractions

I have an equation of the form $\sum_{i=1}^{m}{\frac{1}{a_{i}-x}}=\sum_{j=1}^{n}{\frac{1}{b_{j}-x}}$ and would like to express $x$ as a an approximate explicit function of the $a_{i},b_{j},m,n$. Have ...

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197 views

### A certain sum with q by the power of binomial (n 2)

Is there a closed form to the following sum: $\sum_{n=0}^{\infty}a^nq^{n(n-1)/2}$
for all $a>0$ and $0\lt q\lt 1$ ?

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**3**answers

792 views

### Simplifying finite sum over 1/(ax+b)

Can I simplify:
\begin{equation}
\sum_{x=x_0}^{x_1} \frac{1}{ax+b}
\end{equation}

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394 views

### New results on Chow's notion of closed-form numbers?

In an interesting article (available here), Timothy Chow proposes that a closed-form number be defined as an element of the smallest subfield of $\mathbb{C}$ that is closed under $\exp$ and a chosen ...

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759 views

### Expectation of a simple function of multivariate gaussians iid rvs

I would like to compute analytically the following expected value:
$$ E\left( \frac{X_i^2}{\sum_j \lambda_j^2 X_j^2}\right) $$
where the $X_i \approx N(0,1)$ are iid.
It seems to be an elementary ...

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**1**answer

603 views

### Sum of subset of geometric series: a^2^n

The formula for 1 + a + a^2 + .... where 0 < a < 1 is $\frac{1}{1-a}$, but what if you wanted to sum only those where the exponent is a power of 2? That is,
$S = a + a^2 + a^4 + a^8 + \cdots$
...