Questions tagged [closed-form-expressions]

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

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Finding closed form roots for pseudo-trinomial

I have the below function: $$\pi(x) = \frac{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot r_1}{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) ...
Arthur's user avatar
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Is it possible to evaluate $ \int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{1+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}? $ [closed]

People suggested me to upload this question on math overflow and many people gave the numerical results by desmos and wolfram alpha Motivation for the problem here: Is it possible to evaluate $$ \...
Sbsty 's user avatar
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Is there a closed form of $\int\frac{x\ln x}{\arctan x}dx$?

Consider the integral$$\int\dfrac{x\ln x}{\arctan x}dx\label1\tag1$$While I have made some progress on computing the antiderivative, I would like to know whether or not there is a closed form of it. ...
CrSb0001's user avatar
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What are the terms given by $\int_0^1\int_0^1 x^py^q \sin(\pi xy) (xy)^{xy} (1-xy)^{1-xy} \, dx \, dy$?

For a positive integer $n$ the terms given by \begin{align} & -\int_0^1 x^n \sin(\pi x) x^x (1-x)^{1-x} \, dx \\[8pt] = {} & \int_0^1\int_0^1 (xy)^n \sin(\pi xy) (xy)^{xy} \frac{(1-xy)^{1-xy}}...
Pinteco's user avatar
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7 votes
1 answer
453 views

How to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?

Here in my answer I got the real part for the polylogarithm function at $1+i$ for natural $n$ $$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$ where $$ B_n=\sum_{k=0}^{\...
Faoler's user avatar
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Closed form of the sum of a fast-converging series

The chi-square distribution with $k$ degrees of freedom is $$ f(x)\, dx = \frac1{\Gamma(k/2)} \left( \frac x2\right)^{(k/2)-1} e^{-x/2} \left( \frac{dx} 2 \right) \qquad \text{for } x>0. $$ This ...
Michael Hardy's user avatar
2 votes
1 answer
142 views

$R$-recursion for the A143017

Let $a(n)$ be A143017 i.e. number of $\{2-1-3, 2'^e-31\}$-avoiding permutations of size $n$ (see definition in the Elizalde paper). Here $$ a(n) = \frac{1}{n}\sum\limits_{k=0}^{\left\lfloor\frac{n}{...
Notamathematician's user avatar
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1 answer
246 views

How to evaluate inverse Laplace transform of $e^{- \sqrt{s}} $ using series?

I tried to find an inverse Laplace transform by series as follows $$ f(t)=L^{-1}_s\left(e^{-\sqrt{s}}\right)(t)=L^{-1}_s\left(\sum_{k=0}^{\infty}\frac{(-1)^k}{k!} s^{\frac{k}{2}}\right)(t)$$ and by ...
Faoler's user avatar
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4 votes
1 answer
218 views

Closed-form solution to hyperbolic PDE

Let $A\in C^{\infty}(\mathbb{R}^2)$ be Lebesgue integrable, and $c_1,c_2\in C^{\infty}(\mathbb{R})$ also be Lebesgue integrable. Consider the hyperbolic PDE $$ \begin{cases} \partial_{x,y}u & = A\...
ABIM's user avatar
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1 vote
1 answer
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Analytical solution for a double integral involving logistic functions and Gaussian distributions

I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows: ​$$...
Charles's user avatar
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4 votes
1 answer
182 views

Partition numbers as the specific sums of the A161511

Let $p(n)$ be A000041 i.e. number of partitions of $n$ (the partition numbers). Let $$ \ell(n)=\left\lfloor\log_2 n\right\rfloor $$ Let $a(n)$ be A161511 i.e. number of $1\cdots0$ pairs in the ...
Notamathematician's user avatar
5 votes
1 answer
366 views

Closed-form for the number of partitions of $n$ avoiding the partition $(4,3,1)$

Let $a(n)$ be A309099 i.e. the number of partitions of $n$ avoiding the partition $(4,3,1)$. We say a partition $\alpha$ contains $\mu$ provided that one can delete rows and columns from (the Ferrers ...
Notamathematician's user avatar
2 votes
1 answer
159 views

Exact calculations with Moyal product by "Bopp Shift"

I'm now working on my Phd thesis on the area of deformation quantization and field theory. After doing all the "ground work" (definitions, motivations, basics of the theory etc) I have now ...
Diego Santos's user avatar
3 votes
1 answer
282 views

How can I verify this family of values for hypergeometric functions?

This Wolfram MathWorld page on hypergeometric functions states that An infinite family of rational values for well-poised hypergeometric functions with rational arguments is given by $$_kF_{k-1}\left[...
Sean Svihla's user avatar
5 votes
0 answers
105 views

Ratio of theta functions as roots of polynomials

I already asked the same question here, but received no answer. I did some little progress and so I'm asking again. I was playing with the theta functions with argument $ z = 0 $ $ \vartheta_2(q) =\...
user967210's user avatar
2 votes
0 answers
88 views

Closed form from a slightly modified recursion for transposed Catalan triangle

Let $a_1(n)$ be A000108, i.e. Catalan numbers. Here $$ a_1(n)=\frac{1}{n+1}\binom{2n}{n} $$ Let $a_2(n)$ be A059715, i.e. number of multi-directed animals on the triangular lattice. From OEIS page we ...
Notamathematician's user avatar
2 votes
0 answers
95 views

Ratio of theta function derivatives with theta function

I have the following ratios I want to compute. $$ \frac{ \left( \frac{\partial \vartheta_3(v, q)}{\partial v} \right)^2 }{C + \left(\vartheta_3(v, q)\right)^2 }, $$ where $C$ is a constant. $$ \frac{ \...
CfourPiO's user avatar
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Summation of the following form with non-integer n

I have the following function: $$ G(z) = \sum_{j = 0}^{\infty} \frac{\Gamma(1 + n) z^j}{j ! (\Lambda + jC)^{n+1}} $$ If $n \quad \epsilon \quad \mathbb{Z}^{+} $, the above function can be ...
CfourPiO's user avatar
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2 votes
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Is this integral solvable analytically?

I have this integral that comes from my research with some Fourier Transforms of spectrum functions: $$ G(\tau) = \int_{0}^{\infty} e^{-\Lambda x} x^n e^{i \tau ( c_1 - c_2 e^{-c_3 x} ) } dx $$ where $...
CfourPiO's user avatar
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5 votes
1 answer
174 views

On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?

I. Recurrences In a previous post, it was mentioned how Almkvist-Zudilin did a computer search for solutions to the recurrence relation, $$(n+1)^3s_{n+1}=(2n+1)(an^2+an+b)s_n+c\,n^3s_{n-1}$$ within a ...
Tito Piezas III's user avatar
0 votes
0 answers
98 views

Simplification of summation and reverse search

Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$. Let $b(n)$ be an integer sequence such that $$b(2^m(2n+1))=\sum\limits_{k=0}^{m}f(m-k)b(2^kn), b(0)=1$$ Let $s(n,m)$ be an integer ...
Notamathematician's user avatar
8 votes
2 answers
577 views

On Zagier's missing continued fraction with multiple limits?

I. Zagier's continued fraction As pointed out by Gorodetsky in his answer, Zagier evaluated the continued fractions associated with his six sporadic sequences excepting the one for $(-9,-3,-27)$. Let $...
Tito Piezas III's user avatar
6 votes
1 answer
251 views

On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence

In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
Tito Piezas III's user avatar
10 votes
2 answers
649 views

On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"

I. Some functions As these will be used in the continued fraction evaluations below, recall the Riemann zeta function $\zeta(s),$ and Dirichlet beta function $\beta(s),$ $$\beta(s) = \sum_{n=1}^\infty\...
Tito Piezas III's user avatar
1 vote
0 answers
36 views

Sequences that sum up to $\frac{m^{n+1}}{n+1}\binom{2n}{n}{}_2F_1(1,n+\frac{1}{2}; n+2; -4m(m-1))$

Let $a(n,m)$ be an integer sequence such that $$a(n,m)=\frac{m^{n+1}}{n+1}\binom{2n}{n}{}_2F_1(1,n+\frac{1}{2}; n+2; -4m(m-1))$$ Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ $$f(n)=n-2^{\ell(n)}$$...
Notamathematician's user avatar
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0 answers
34 views

$\frac{m(m+k+1)^n+k}{m+k}$ as closed form for subsequence of the partial sums

Let $a(n,m,k)$ be an integer sequence such that $$a(n,m,k)=\frac{m(m+k+1)^n+k}{m+k}$$ There are many sequences in the OEIS that are special cases of a given sequence family: $a(n,1,1)$ - A007051 $a(n,...
Notamathematician's user avatar
0 votes
0 answers
40 views

Product as closed form for subsequence of the partial sums

Let $a(n,m,k)$ be an integer sequence such that $$a(n,m,k)=\prod\limits_{q=0}^{n-1}\sum\limits_{i=0}^{m-1}\sum\limits_{j=0}^{m-i-1}\binom{i+j-1}{j}k^{i+j}q^i$$ Let $$\ell(n,m)=\left\lfloor\log_m n\...
Notamathematician's user avatar
4 votes
2 answers
495 views

Finding a sextic analogue to the solvable octic $\frac{(x + 1)^6(x^2 + x + 7)}x = -k^3$ where $e^{(\pi/3)\sqrt{d}}\approx k^3+41.999999999999\dots$

I. Degree 8 Assume the $j_i$ to be free parameters. The following octics in $x$ belong to $8T43,$ have group $\text{PGL}(2,7)$, and order $2\times168 = 336.$ \begin{align} {j_1}\; &=\frac{(x^2 + ...
Tito Piezas III's user avatar
0 votes
0 answers
67 views

On a generalization of A113227 as a subsequence of the partial sums

This question is just a generalization of the one of my previous questions. Let $$a(n,m,k)=\sum\limits_{i=1}^{n}u(n,m,k,i)$$ where $$u(n,m,k,i)=u(n-1,m,k,i-1)+(m-1)(i+k-1)\sum\limits_{j=i}^{n-1}u(n-1,...
Notamathematician's user avatar
4 votes
0 answers
85 views

Closed form for subsequence of the partial sums of generalized A329369

Let $a(n,m,k)$ be an integer sequence such that $$a(n,m,k)=\sum\limits_{i=0}^{n}{n\brace i}(m-k)^{n-i}\prod\limits_{j=0}^{i-1}(kj+1)$$ Here ${n\brace k}$ is the Stirling number of the second kind. ...
Notamathematician's user avatar
1 vote
1 answer
193 views

Analytic expression for $\int_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\phi ))}{p \left(e^{c p}+1\right)}$

I am looking for ways to do this integration analytically \begin{equation} \int_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\phi ))}{p \left(e^{c p}+1\right)} \end{equation} For ...
user824530's user avatar
17 votes
1 answer
832 views

On the solvable septic quadrinomial $x^7-7x^4-14x^3-7=0$?

The concept of sparse polynomials has its place, and solvable but irreducible quadrinomial examples such as, $$x^7-7x^4-14x^3-7=0$$ $$x^8+x^7+29x^2+29=0$$ $$x^9-27x^4-9x^3-9^2=0$$ $$x^{12}-36x^5-12x^3-...
Tito Piezas III's user avatar
1 vote
0 answers
84 views

Derive a closed-form expression of this recursive formula

$$\begin{equation} S(r,k) = f(r)S(0,k-1) + g(r)S(r+1,k-1) \end{equation}\ ,$$ where $r=0,1,2,\dots$ and $k=1,2,3,\dots$ . Also, $0<f(r)<1$ is an increasing function and $0<g(r)<1$ is a ...
K. Bountrogiannis's user avatar
1 vote
1 answer
110 views

Coefficients of number of the same terms which are arising from iterations based on binary expansion of $n$

Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ Let $$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$ Here $T(n,k)$ is the $(k+1)$-th bit from the right side in the binary ...
Notamathematician's user avatar
1 vote
0 answers
119 views

$\sin(\frac{\pi}{p}) $ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$?

We have the following identities: $\sin(\frac{\pi}{1})=0$ $\sin(\frac{\pi}{2})=1$ $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$ $\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$ Lets start with a definition. Rules ...
mick's user avatar
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3 votes
1 answer
250 views

Integrating $\int_0^\infty \sqrt x e^{-4/3x^{3/2}}\left(\int_0^x \operatorname{Ai}(t)dt\int_0^x \operatorname{Bi}(t)dt\right)dx$

Show that $$I= \int_0^\infty \sqrt x e^{\large -4/3x^{3/2}}\left(\int_0^x \operatorname{Ai}(t)dt\int_0^x \operatorname{Bi}(t)dt\right)dx$$ $$=\frac{1}{3}-\frac{\sqrt[3]{2\sqrt 3+3}+\sqrt[3]{2\sqrt 3-3}...
Zacky's user avatar
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3 votes
1 answer
353 views

Using the Lehmer quintic to solve $11$-degree equations and higher?

(This is a natural continuation of a previous post.) I. Quintic method Given the Lehmer quintic, $$x^5 + n^2x^4 - (2n^3 + 6n^2 + 10n + 10)x^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)x^2 + (n^3 + 4n^2 + 10n + ...
Tito Piezas III's user avatar
0 votes
0 answers
72 views

How to perform this integral to get a closed/ semi closed form

I want to get a closed/ semi-closed form of the integral given below. $$ \int_{-\infty}^{+\infty} \exp{\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)} \text{erf}\left(\alpha \frac{x-\mu}{\sqrt{2}\sigma}\...
CfourPiO's user avatar
  • 159
-2 votes
1 answer
168 views

Simple closed form for $\int \lfloor x \rfloor dx$? [closed]

Wolfram Alpha claims there is no closed form in terms of standard funcions for $\int \lfloor x \rfloor dx$ but we believe we found simple closed form agreeing with experimental data. Define $i_1(x)=x -...
joro's user avatar
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7 votes
2 answers
411 views

A method to generate solvable equations of degrees $p = 7, 13, 19, 31, 37,\dots$ using only cubics

I've always wondered if the DeMoivre method to generate an algebraic number $x_p$, $$x_p = u_1^{1/p}+u_2^{1/p}$$ of degree $p$ using only quadratic roots $u_i$ could be generalized using cubic roots $...
Tito Piezas III's user avatar
2 votes
0 answers
157 views

Closed form for the A347205

Let $q(n)$ be A007814, i.e., number of trailing zeros in the binary representation of $n$. Here $$q(2n+1)=0, q(2n)=q(n)+1$$ Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary ...
Notamathematician's user avatar
2 votes
0 answers
112 views

Closed form for the sum of the integer coefficients

Let $a(n)$ be A002720, i.e., number of partial permutations of an $n$-set; number of $n \times n$ binary matrices with at most one $1$ in each row and column. $$a(n)=\sum\limits_{k=0}^{n} k!\binom{n}{...
Notamathematician's user avatar
5 votes
2 answers
1k views

Integral of a product between two normal distributions and a monomial

The integral of the product of two normal distribution densities can be exactly solved, as shown here for example. I'm interested in compute the following integral (for a generic $n \in \mathbb{N}$): $...
user1172131's user avatar
3 votes
1 answer
128 views

Convolution between normal distribution and the maximum over $m$ Gaussian draws

$\DeclareMathOperator\erf{erf}$ Let's consider the Gaussian distribution $P_X(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{x^2}{2 \sigma^2}}$. Now consider the random variable $W \equiv \max \{ X_1, ...
user1172131's user avatar
2 votes
0 answers
68 views

Closed form for the number of permutations with a given excedance set

Let $q(n)$ be A007814, i.e., number of trailing zeros in the binary representation of $n$. Here $$q(2n+1)=0, q(2n)=q(n)+1$$ Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary ...
Notamathematician's user avatar
0 votes
0 answers
92 views

Closed form for the number of steps required to get $n$ balls in the last box

Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expansion of $n$ (or the binary weight of $n$). Then we have an integer sequence given by $$a(n)=n(n+1)-\sum\limits_{k=0}^{n}\...
Notamathematician's user avatar
2 votes
0 answers
56 views

any ideas on how to solve matrix equation like this $X A_i Y = B_i$

the objective function is like $$\operatorname*{argmin}_{X,Y} = \sum_i \|X A_i Y - B_i\|_F^2$$, and $A_i$ is a diagonal matrix I've tried gradient-descent, but as it turns out not well, I wonder if ...
Cup Y's user avatar
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2 votes
0 answers
115 views

Closed form for coefficients related to excedance set of permutation

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,...
Notamathematician's user avatar
3 votes
0 answers
162 views

Closed form for $a(2^m(2^n-2^p-1))$

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(...
Notamathematician's user avatar
4 votes
1 answer
185 views

Closed-form examples of CMC surfaces

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 ...
Jiří Minarčík's user avatar

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