Questions tagged [closed-form-expressions]
For questions that specifically ask for determining a closed form of equations, integrals etc.
198
questions
2
votes
1
answer
102
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 ...
- 470
3
votes
1
answer
271
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 $$_kF_{k-1}\left[...
- 33
5
votes
0
answers
86
views
Ratio of theta functions as roots of polynomials
I already asked the same question here, but received no answer. I did some little progress and so I'm asking again.
I was playing with the theta functions with argument $ z = 0 $
$ \vartheta_2(q) =\...
- 191
2
votes
0
answers
88
views
Closed form from a slightly modified recursion for transposed Catalan triangle
Let $a_1(n)$ be A000108, i.e. Catalan numbers. Here
$$
a_1(n)=\frac{1}{n+1}\binom{2n}{n}
$$
Let $a_2(n)$ be A059715, i.e. number of multi-directed animals on the triangular lattice. From OEIS page we ...
- 2,886
2
votes
0
answers
87
views
Ratio of theta function derivatives with theta function
I have the following ratios I want to compute.
$$ \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{ \...
- 159
0
votes
0
answers
36
views
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 ...
- 159
2
votes
1
answer
156
views
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 $...
- 159
4
votes
1
answer
164
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 ...
- 11.7k
0
votes
0
answers
97
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 ...
- 2,886
8
votes
2
answers
558
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 $...
- 11.7k
6
votes
1
answer
241
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-...
- 11.7k
10
votes
2
answers
620
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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 $\beta(s),$
$$\beta(s) = \sum_{n=1}^\infty\...
- 11.7k
1
vote
0
answers
36
views
Sequences that sum up to $\frac{m^{n+1}}{n+1}\binom{2n}{n}{}_2F_1(1,n+\frac{1}{2}; n+2; -4m(m-1))$
Let $a(n,m)$ be an integer sequence such that
$$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)}$$...
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0
votes
0
answers
34
views
$\frac{m(m+k+1)^n+k}{m+k}$ as closed form for subsequence of the partial sums
Let $a(n,m,k)$ be an integer sequence such that
$$a(n,m,k)=\frac{m(m+k+1)^n+k}{m+k}$$
There are many sequences in the OEIS that are special cases of a given sequence family:
$a(n,1,1)$ - A007051
$a(n,...
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0
votes
0
answers
40
views
Product as closed form for subsequence of the partial sums
Let $a(n,m,k)$ be an integer sequence such that
$$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\...
- 2,886
4
votes
2
answers
490
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 + ...
- 11.7k
0
votes
0
answers
64
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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
$$u(n,m,k,i)=u(n-1,m,k,i-1)+(m-1)(i+k-1)\sum\limits_{j=i}^{n-1}u(n-1,...
- 2,886
4
votes
0
answers
80
views
Closed form for subsequence of the partial sums of generalized A329369
Let $a(n,m,k)$ be an integer sequence such that
$$a(n,m,k)=\sum\limits_{i=0}^{n}{n\brace i}(m-k)^{n-i}\prod\limits_{j=0}^{i-1}(kj+1)$$
Here ${n\brace k}$ is the Stirling number of the second kind.
...
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1
vote
1
answer
169
views
Analytic expression for $\int_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\phi ))}{p \left(e^{c p}+1\right)}$
I am looking for ways to do this integration analytically
\begin{equation}
\int_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\phi ))}{p \left(e^{c p}+1\right)}
\end{equation}
For ...
- 199
17
votes
1
answer
787
views
On the solvable septic quadrinomial $x^7-7x^4-14x^3-7=0$?
The concept of sparse polynomials has its place, and solvable but irreducible quadrinomial examples such as,
$$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-...
- 11.7k
1
vote
0
answers
81
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 ...
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 ...
- 2,886
1
vote
0
answers
116
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 ...
- 677
3
votes
1
answer
239
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
$$I= \int_0^\infty \sqrt x e^{\large -4/3x^{3/2}}\left(\int_0^x \operatorname{Ai}(t)dt\int_0^x \operatorname{Bi}(t)dt\right)dx$$
$$=\frac{1}{3}-\frac{\sqrt[3]{2\sqrt 3+3}+\sqrt[3]{2\sqrt 3-3}...
- 175
3
votes
1
answer
347
views
Using the Lehmer quintic to solve $11$-degree equations and higher?
(This is a natural continuation of a previous post.)
I. Quintic method
Given the Lehmer quintic,
$$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 + ...
- 11.7k
0
votes
0
answers
68
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.
$$ \int_{-\infty}^{+\infty} \exp{\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)} \text{erf}\left(\alpha \frac{x-\mu}{\sqrt{2}\sigma}\...
- 159
-2
votes
1
answer
167
views
Simple closed form for $\int \lfloor x \rfloor dx$? [closed]
Wolfram Alpha claims there is no closed form in terms of standard funcions
for $\int \lfloor x \rfloor dx$ but we believe we found
simple closed form agreeing with experimental data.
Define $i_1(x)=x -...
- 24k
7
votes
2
answers
404
views
A method to generate solvable equations of degrees $p = 7, 13, 19, 31, 37,\dots$ using only cubics
I've always wondered if the DeMoivre method to generate an algebraic number $x_p$,
$$x_p = u_1^{1/p}+u_2^{1/p}$$
of degree $p$ using only quadratic roots $u_i$ could be generalized using cubic roots $...
- 11.7k
0
votes
0
answers
27
views
Distribution of the second bigger value among m Gaussian draws
$\DeclareMathOperator\erf{erf}$
Let's consider the set $\Omega_m$ of $m$ i.i.d. Gaussian variables $\{X_1, X_2, \dots, X_m \}$, with $X_i \sim \mathcal{N} \left( 0, \sigma^2 \right) \forall i$.
$\...
- 155
2
votes
0
answers
154
views
Closed form for the A347205
Let $q(n)$ be A007814, i.e., number of trailing zeros in the binary representation of $n$. Here
$$q(2n+1)=0, q(2n)=q(n)+1$$
Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary ...
- 2,886
2
votes
0
answers
111
views
Closed form for the sum of the integer coefficients
Let $a(n)$ be A002720, i.e., number of partial permutations of an $n$-set; number of $n \times n$ binary matrices with at most one $1$ in each row and column.
$$a(n)=\sum\limits_{k=0}^{n} k!\binom{n}{...
- 2,886
4
votes
2
answers
778
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 $n \in \mathbb{N}$):
$...
- 155
3
votes
1
answer
106
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, ...
- 155
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 ...
- 2,886
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
$$a(n)=n(n+1)-\sum\limits_{k=0}^{n}\...
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0
votes
0
answers
22
views
Families of 1st order recursive relations with closed forms
A general 1st order recurrence relation can be expressed as $X_{n+1} = f(X_{n},n)$.
I am looking for a source, list of functions or statement about $f$ such that an $F$ exists which expresses $X_{n} = ...
- 139
0
votes
0
answers
64
views
Examples of $f(n,m)$-solvable recurrences
Let $a_k(n)$ be an integer sequence given by recurrence relation
$$a_k(2n+1)=a_k(n), a_k(2n)=a_k(n)+\sum\limits_{j=1}^{i}a_k(g_j(n)), a_k(0)=1$$
Here $g_j(n)$ are some functions in most cases based on ...
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2
votes
0
answers
56
views
any ideas on how to solve matrix equation like this $X A_i Y = B_i$
the objective function is like
$$\operatorname*{argmin}_{X,Y} = \sum_i \|X A_i Y - B_i\|_F^2$$, and $A_i$ is a diagonal matrix
I've tried gradient-descent, but as it turns out not well, I wonder if ...
- 21
2
votes
0
answers
115
views
Closed form for coefficients related to excedance set of permutation
Working on suitable closed form for A329369, I discovered very useful coefficients, which have the following recurrence relation:
$$T(0,1)=T(0,2)=1$$
$$T(n,1)=1, n>0$$
$$T(0,k)=0, k>2$$
$$T(2n+1,...
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3
votes
0
answers
159
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
$$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(...
- 2,886
4
votes
1
answer
180
views
Closed-form examples of CMC surfaces
Besides the trivial cases of cylinders and spheres, are there any other known examples of non-zero constant mean curvature surfaces which can be represented explicitly in a closed form? I am ...
- 151
1
vote
1
answer
105
views
Number of steps required to get one ball in each box for $n=2^k$
Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every ...
- 2,886
3
votes
1
answer
497
views
Is there real or complex analytic function whose positive real zeros are the primes?
Related to this question
Q1 Is there real or complex analytic function $f(x)$ such
that its positive real zeros are the primes and it is
given in closed form of compositions of already named ...
- 24k
9
votes
1
answer
728
views
Closed form for ₄F₃(n,n,n,2n;1+n,1+n,1+n;−1)
For positive integer $n$ the following value of a hypergeometric function
$$_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$$
where $R_{1,2}(n)$ are ...
- 986
1
vote
0
answers
164
views
Solution that minimizes the sum of squared errors, with quadratic constraints
Given symmetric and positive definite $n \times n$ (real) matrices $A_1, \dots, A_m$ and $b_1, \dots, b_m \in {\Bbb R}^{n}$, I am trying to find the solution with the least sum of squared errors of ...
- 11
2
votes
0
answers
64
views
How to extend this sum involving generalized harmonic numbers?
It is well-known since Euler that the Generalized harmonic numbers, defined for $n\in\mathbb N$ by $$H_n^{(r)}=\sum_{k=1}^n\frac1{k^r},$$ can be naturally extended for non integer $n$ in terms of ...
- 13.1k
2
votes
1
answer
143
views
How to solve for $a$ in $\sum_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j} = 0$
The Problem
Given $i, n, y$, I am trying to find a closed form solution for $a$ in the equation
$$
\sum_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j}
=
0
$$
Do you have any recommendations on how to ...
- 23
1
vote
1
answer
296
views
Is this long closed form for pi trivial?
With the help of wolfram alpha we got very long closed form
for $\pi$ in terms of algebraic numbers, logarithms of algebraic
numbers and $cot^{-1},coth^{-1}$ which can be expressed as logarithms.
From ...
- 24k
9
votes
3
answers
508
views
Why mpmath computes $\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right)$
Working with precision 500 decimal digits, mpmath in sage computes:
$$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$
We believe the LHS of \eqref{1} ...
- 24k
4
votes
3
answers
1k
views
Surprisingly long closed form for simple series
For natural $A$ define
$$ 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 $A$.
...
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