# Questions tagged [closed-form-expressions]

For questions that specifically ask for determining a closed form of equations, integrals etc.

For questions that specifically ask for determining a closed form of equations, integrals etc.

162
questions

2
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0
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74
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Working on suitable closed form for A329369, I discovered very useful coefficients, which have the following recurrence relation:
$$T(0,1)=T(0,2)=1$$
$$T(n,1)=1, n>0$$
$$T(0,k)=0, k>2$$
$$T(2n+1,...

2
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0
answers

90
views

Let $q(n)$ be A007814, i.e., the number of trailing zeros in the binary representation of $n$. Here
$$q(2n+1)=0, q(2n)=q(n)+1$$
Let $a(n)$ be A329369. Here
$$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(...

4
votes

1
answer

153
views

Besides the trivial cases of cylinders and spheres, are there any other known examples of non-zero constant mean curvature surfaces which can be represented explicitly in a closed form? I am ...

1
vote

1
answer

96
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Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every ...

3
votes

1
answer

425
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Related to this question
Q1 Is there real or complex analytic function $f(x)$ such
that its positive real zeros are the primes and it is
given in closed form of compositions of already named ...

8
votes

1
answer

704
views

For positive integer $n$ the following value of a hypergeometric function
$$_4F_3(n,n,n,2n,1+n,1+n,1+n,-1)$$
based on the first few terms looks like
$$ R_1(n) + R_2(n) \pi^2$$
where $R_{1,2}(n)$ are ...

1
vote

0
answers

114
views

Given symmetric and positive definite $n \times n$ (real) matrices $A_1, \dots, A_m$ and $b_1, \dots, b_m \in {\Bbb R}^{n}$, I am trying to find the solution with the least sum of squared errors of ...

1
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0
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49
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It is well-known since Euler that the Generalized harmonic numbers, defined for $n\in\mathbb N$ by $$H_n^{(r)}=\sum_{k=1}^n\frac1{k^r},$$ can be naturally extended for non integer $n$ in terms of ...

1
vote

1
answer

115
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The Problem
Given $i, n, y$, I am trying to find a closed form solution for $a$ in the equation
$$
\sum_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j}
=
0
$$
Do you have any recommendations on how to ...

3
votes

0
answers

119
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Let's define
$$\sigma(m,n)=\sum_{k=1}^\infty\frac{H_k^{(m)}}{k^n}$$
where $H_k^{(m)}=\sum_{n=1}^{k}\frac{1}{n^m}$ is the k-th generalized harmonic number of order $m$.
In mathworld site Eq (20), I ...

1
vote

1
answer

257
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With the help of wolfram alpha we got very long closed form
for $\pi$ in terms of algebraic numbers, logarithms of algebraic
numbers and $cot^{-1},coth^{-1}$ which can be expressed as logarithms.
From ...

8
votes

3
answers

472
views

Working with precision 500 decimal digits, mpmath in sage computes:
$$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$
We believe the LHS of \eqref{1} ...

3
votes

3
answers

1k
views

For natural $A$ define
$$ f(A)=\sum_{n=1}^\infty \frac{1}{A^n}\left(\frac{1}{An+1}- \frac{1}{An+A-1}\right)$$
$f(A)$ is BBP (Bailey-Borwein-Plouffe) formula and allows digit extraction in base $A$.
...

0
votes

0
answers

29
views

I need to solve for $D$:
$$
KD^{N-1} S + P = KD^N + \left(\frac{D}{N}\right)^N,
$$
where
$$
S = \sum_i x_i, \, P = \prod_i x_i,
$$
$$
K_0 = P \left(\frac{N}{D}\right)^N, \, K = AK_0 \frac{\gamma^2}{(\...

3
votes

0
answers

77
views

Consider the following symmetric, positive-definite matrix
$$
H_{nm}=-\frac{(4 m+4 n+1)}{(4 m-4 n-1) (4 m-4 n+1)}\sqrt{\frac{(4 m-1)!! (4 n-1)!!}{(4 m)!! (4 n)!!}}
$$
where $n,m=0,1,2\dots$ (Here $!!$ ...

5
votes

1
answer

157
views

I am interested in the asymptotics of the integral
$$I(a):=\int_0^\infty \sqrt{x}\operatorname{Erfc}(x+a)\,\mathrm{d}x$$
for $a>0$. I think that $I(a)$ should be decaying exponentially as $I(a)\...

4
votes

0
answers

132
views

I am interested in the sum
$$
\sum_{n=1}^k 2\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g}
$$
where $k$, $g$ are integers. It is not too hard to show that this can also be expressed as
$$
-1-...

2
votes

0
answers

154
views

I have a pretty simple question for which I was not able to find a so simple answer.
Introduction
I was playing around with some of the mathematical objects that can be enumerated by Catalan numbers. ...

0
votes

0
answers

57
views

Let $\Delta_n:=\{x\in [0,1]^n:\boldsymbol{1}^{\top}x=1\}$ denote the probabilist's $n$-simplex and let $P:\mathbb{R}^n\rightarrow\Delta_n$ denote the (Euclidean) metric projection onto this simplex ...

1
vote

1
answer

75
views

Here is the goal sum where the Pochhammer Symbol with the Incomplete Beta function series
$$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...

3
votes

1
answer

177
views

The following integrals are equal to the volume of a unit ball in $S^2 \times \mathbb R$ and $H^2 \times\mathbb R$, respectively:
$$8\pi\int_0^1\sin^2 \frac{\sqrt{1-h^2}}2 \, dh$$
$$8\pi\int_0^1\sinh^...

1
vote

0
answers

110
views

I am looking for the closed form of
$$\sum _{k=1} ^n \frac k 2 \,\operatorname{sgn} \left( \frac 1 {k^2} + \cos \frac {2\pi n} k-1 \right) \ .$$
Wolfram Alpha cannot do this for me, so I am forced to ...

2
votes

0
answers

33
views

The Lauricella function $F_B^{n}$ of $n$ variables is defined as $$F_B^{(n)}(a_1, \ldots, a_n, b_1, \ldots, b_n, c; x_1, \ldots x_n) = \sum_{k_1, \ldots, k_n = 0}^\infty \frac{1}{(c)_{k_1 + \ldots + ...

0
votes

1
answer

191
views

The infinite series representation
\begin{equation}
\frac1\pi\sum_{j=0}^\infty \biggl(j+\frac{1}{2}\biggr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}, \quad n\ge0
\end{equation}
for the positive ...

2
votes

0
answers

215
views

where $\phi = \frac{1+\sqrt{5}}{2}$ and $k!_F$ is the fibonorial of $k$, or the product of the first $k$ Fibonacci numbers? My hunch is that, this can be represented as a function in terms of the ...

1
vote

0
answers

158
views

Starting with the following differential equation,
\begin{eqnarray}
x \frac{\partial^3}{\partial x^3} P[h, x] - \frac{\partial^2}{\partial h^2} \left( h \frac{\partial}{\partial h} P [h, x] \right) ...

1
vote

0
answers

83
views

Two years back I was working with lacunary series. In my explorations I had derived that the following series is periodic with period 1:
$$ f(x)= \sum_{n=0}^{\infty} \left[ e^{2^{x+n}} \right] + x - \...

1
vote

0
answers

29
views

I have the following dynamic programming principle-type problem.
Suppose that we are given a sequence $\beta_1,\dots,\beta_n\in (0,\infty)$, some target $y\in (0,\infty)$ with $y>\sum_{t=1}^N \...

0
votes

0
answers

166
views

Consider the Series
$$
S(z) \equiv \sum_{n \in \mathbb{Z}, n \ne 0} \frac{ 1 }{ \sin n\pi \tau \sin 2n \pi \tau } e^{2\pi i n z} \ , \quad \operatorname{Im}\tau > 0
$$
I am trying to find its ...

1
vote

1
answer

216
views

Consider the summatory function $f_a(x)$ defined in formula (1) below where the related Dirichlet series $F_a(s)$ defined in formula (2) below converges for $\Re(s)\ge 2$.
$$f_a(x)=\sum\limits_{n=1}^...

0
votes

0
answers

89
views

$$ \sum_{n=0}^{k+1}\frac{3F_{n+1}-L_{n+1}}{2n!}\frac{(k+1)!}{(k-n+1)!}x^{k-n+1}=(\varphi+x)^k\left(\frac{\sqrt{5}}{5}-\frac{\sqrt{5}-5}{10}x\right)+(\psi+x)^k\left(\frac{\sqrt{5}+5}{10}x-\frac{\sqrt{5}...

2
votes

1
answer

280
views

Is there any closed form of
$$\prod_{k=1}^{n}\left(\cos(kx)-1\right)?$$
I failed to find references on this problem in the internet.

3
votes

2
answers

359
views

NOTE: I post this question on math.stackexchange but nobody answered, so I try here.
For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-...

3
votes

0
answers

199
views

Let $f$ be an even continuous function on the sphere $S^{n-1}$.
Find a relation of the spherical harmonic expansion between coefficients of $f^n$ and those of $f$.

1
vote

0
answers

84
views

Let $ \mathbb{F} $ be a field, consider the polynomial ring $ \mathbb{F} \left[ x\right] $ and suppose that the polynomials $f,g \in \mathbb{F} \left[ x\right]$ have degrees $2,2^n$, respectively, ...

3
votes

1
answer

76
views

Question. Is there a closed-form formula (via standard objects like rational functions, radicals, special functions, special numbers like Catalan numbers, etc.) expressing integrals of rational ...

0
votes

2
answers

255
views

Let $a$ be an element of some ring or field, possibly finite.
Is there closed form for $\sum_{i=1}^n{a^{i^2}}$?
sage and wolframalpha couldn't solve it.
If $a$ is primitive n-th root of unity this is ...

6
votes

4
answers

395
views

With some Poisson summation manipulations (credit: Michał Pacholski) I have convinced myself of a closed form expression for this conditionally convergent series:
$$\sum_{n=-\infty}^\infty \frac{e^{in\...

7
votes

0
answers

260
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Note: This question has been brought here from MSE.
I have been working on various sums involving the zeta function (which come up frequently in my research), and it turned out that many of them had ...

0
votes

1
answer

96
views

I asked this question of MSE, but to no avail; alas, here I am.
Let $k>0$, $C\geq 1$, $\alpha \in (0,1]$, and let $(x_n)_{n\geq 1}$, be a sequence of real numbers given by the recursion
$$
x_{n+1} =...

1
vote

1
answer

119
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I asked how to calculate $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$ and got amazing answers. A bit later, however, I figured I needed something rather more complicated: I need to find ...

-1
votes

3
answers

272
views

Any idea on whether or not $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$
has a closed formula on $a$ and $b$ (and on what it is, in case it does)?
It is supposed that $b \le a$.

5
votes

2
answers

407
views

As stated, I wonder if there is a closed form for the generating function $F_{\alpha,\beta}(x):=\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$ where $\alpha,\beta \in\mathbb{N}$. Calling ...

2
votes

0
answers

104
views

I want to solve the following optimization problem in variables $\theta_1, \theta_2, \dots, \theta_K$
\begin{equation}
\begin{aligned}
& \underset{\theta}{\text{minimize}}
& & \...

8
votes

3
answers

391
views

This is my first post. I'm not a mathematician, just an electronics engineer who loves mathematics. In one of my projects, I arrived at the following function:
$$V\left(\varphi\right)=\frac{A\sqrt{\pi-...

6
votes

0
answers

129
views

We have no idea how to tackle the following Fourier transformation of a distribution:
$$
\lim_{\epsilon\to0^+}\int_{-\infty}^{\infty}\mathrm{d} t\int_{\mathbb{R}^{d-1}} \mathrm{d}^{d-1}\vec{r} e^{-\...

1
vote

1
answer

191
views

Given $a,b,c,d\in\mathbb Z$ with $ad+bc=p$ a prime there is an $m\in\mathbb Z$ with $-\lceil1+\sqrt{p}\rceil<
r_1,r_2<\lceil1+\sqrt{p}\rceil$ and
$$r_1\equiv mac\bmod p$$
$$r_2\equiv mbd\bmod p$$...

4
votes

0
answers

165
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I'm interested in sums of the form $$f_{p} (x) = \sum_{k=2}^{\infty} \zeta(k)^{p} x^{k} .$$
For $p=1$, the following result is known: $$f_{1} (x) = -x \big{(}\psi(1-x) + \gamma \big{)} .$$
(That is, ...

-2
votes

1
answer

210
views

let $\epsilon >0$, I tried to evaluate $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ , using the fact $x= \cos t$ yield to have integrand using $\sin $ ...

1
vote

0
answers

90
views

Is my conjecture below true?
It's a new conjecture. It's an extension of Lin's theorem in [Lin 1983] which is cited in [Chow 1999] - see both references at the bottom of this page.
It seems that Ferng-...