Questions tagged [closed-form]

A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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4
votes
1answer
38 views

Closed form for $\sum_{n=1}^{\infty}\frac{x^n}{n!\sqrt{n}}$, or an asymptotic for it [duplicate]

In my study I came about the function defined by $$f(x)=\sum_{n=1}^{\infty}\frac{x^n}{n!\sqrt{n}}$$ and I have not been able to find any sort of closed form for it, nor have I been able to find any ...
1
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0answers
29 views

Closed-Form solution for nested integrals of this polynomial?

I was wondering whether there is a closed-form solution for this (nested) integral: $$ \int_{-1}^{1}\int_{t_{0}}^{1}\int_{t_{1}}^{1}...\int_{t_{a-2}}^{1}\prod_{\begin{array}{c} i<j\\ j=\{0,..,a-1\}\...
1
vote
1answer
55 views

Closed Form Solution to System of Equations

Let $ \mathbf{b} \in \mathbb{R}_+^n$, $\mathbf{V} \in \mathbb{R}_+^{n \times m}$ with $\mathbf{V} \mathbf{1}_m = \mathbf{1}_n$ where $\mathbf{1}_n$ is the vector of ones of size $n$. I have the ...
2
votes
1answer
88 views

Beautiful closed form of $\sum_{n=1}^{\infty}\frac{(-1)^n}{2n^2}\int_0^1\left(\ln f(x)+2\ln g(x)\right)dx$

While solving the following the integral which I found here in brilliant $$\int_0^1\left(\ln(4-3^x)+\ln(1+3^x)\right)dx$$ I happen to create the general integral and variant for the aforementioned ...
7
votes
3answers
74 views

Closed form sought for $a_1 = a_2 = 1, a_n = 1 + \frac{2}{n} \sum_{i=1}^{n-2} a_i $ where $n>2$

I've been working through a problem that I've got as far as getting a recursive answer to. I was hoping to turn this into more of a "closed form" answer, but haven't really gotten anywhere. ...
5
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1answer
48 views

Proving a pair of hypergeometric identities

Recently I have computed several classes of high weight hypergeometric series. While surfing on this site, I found only 2 interesting identities of high weight which are not deducible from basic ...
1
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2answers
64 views

A formula for $D(x)D(y) - D(xy)$ in terms of the sum-of-aliquot-divisors function, when $\gcd(x,y)=1$

Hereinafter, we shall let $\sigma(z)$ be the sum of divisors of the positive integer $z$. Denote the deficiency of $z$ by $D(z) = 2z - \sigma(z)$, and the sum of aliquot divisors of $z$ by $s(z) = \...
0
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1answer
32 views

Recursive to explicit form involving Fibonacci

I have a recursive formula for a sequence O: $ O_n = O_{n-1} + O_{n-2} + F_{n-1}$ where $F_n$ is the n-th Fibonacci number, $O_1 = 1$ and $O_2 = 2$. After playing around with it, I found a new formula ...
0
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0answers
51 views

Is there a nice way to represent $\prod_{m=1}^{2^{n-1}} \cos\left(2^{m-1} x\right)$?

Does $$\prod_{m=1}^{2^{n-1}} \cos\left(2^{m-1} x\right) \stackrel{?}{=} \frac{\sin\left(2^nx\right)}{2^n\sin x}?$$ I want a more rigorous way to derive this than my approach: Using that $\sin(2x)= 2\...
0
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0answers
25 views

Closed-form of a Gaussian and a Bessel-like integral

I am trying to solve the following integral $$ I=\int_0^{\infty}dx_1 dx_2 dx_3 dx_4 \times x_1 x_2 x_3 x_4 e^{-a_1 x_1^2-a_2x_2^2-a_3x_3^2-a_4x^4} I_{m_1}(c_1 x_1 x_2) I_{m_2}(c_2 x_1 x_3) I_{m_3}(c_3 ...
3
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2answers
61 views

Closed-form solution for the determinant of a Vandermonde-like matrix

I'm trying to find a closed-form solution $\forall$ odd integer $n\ge 3$ for the determinant of a matrix with some structure on it. After some manipulation, I've reduced it to the following matrix: $\...
1
vote
1answer
81 views

Closed form of $\sum_{k=1}^{n} \frac{{n}\choose{k}}{k}$.

I would like to ask if it is possible to find a closed form of the sum $\sum_{k=1}^{n} \frac{{n}\choose{k}}{k}$ (1). I managed to show that it is enough to find a closed form of $\sum_{k=1}^{n} \frac{...
4
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3answers
120 views

Proving $\sum_{n=0}^\infty\frac{(-1)^n\Gamma(2n+a+1)}{\Gamma(2n+2)}=2^{-a/2}\Gamma(a)\sin(\frac{\pi}{4}a)$

Mathematica gives $$\sum_{n=0}^\infty\frac{(-1)^n\Gamma(2n+a+1)}{\Gamma(2n+2)}=2^{-a/2}\Gamma(a)\sin(\frac{\pi}{4}a),\quad 0<a<1$$ All I did is reindexing then using the series property $\sum_{n=...
2
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3answers
92 views

Closed form expression for $\sum\limits_{k=0}^{\infty}\frac{k4^k}{n+k}{n+k\choose n-k}{n\choose k}{2n\choose 2k}^{-1}$

I'm trying to find a closed form expression (that doesn't involve an indefinite summation) for the following combinatorial sum : $$\sum\limits_{k=0}^{n}\left[\frac{k4^k}{n+k}{n+k\choose n-k}\frac{\...
2
votes
3answers
71 views

Any nice (not necessarily closed) forms for ${\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\eta(2n)}{n}}$?

I've been playing around with series involving the eta function and I (think) managed to find a nice form for the series in the title (not a closed form, but a form with a very nice pattern). The ...
1
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0answers
111 views
+100

The conjecture about the existence of closed-form inverses of functions

Is my proof draft below already a proof? How can the proof be completed? Definition: A unary complex function is a function from a subset of $\mathbb{C}$ into $\mathbb{C}$. A binary complex function ...
5
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2answers
111 views

Alternative approaches to showing that $\Gamma'(1/2)=-\sqrt\pi\left(\gamma+\log(4)\right)$

Starting from the definition of the Gamma function as expressed by $$\Gamma(z)=\int_0^\infty x^{z-1}e^{-x}\,dx\tag1$$ we can show that the derivative of $\Gamma(z)$ evaluated at $z=1/2$ is given by $$\...
1
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0answers
39 views

Closed form of $\sum_{n=1}^\infty (n+k)!(a/n)^n$

I got this equality: $$\sum_{n=1}^\infty (n+k)!\left(\frac{a}{n}\right)^n=a(k+1)!\int_{0}^{1}\frac{dx}{(1+ax\ln x)^{k+2}}$$ when $|a|<e$ then, does this series have a closed form?
5
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1answer
123 views

Evaluate $_6F_5\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1,1;\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2};1\right)$

Background: I'm looking for hypergeometric representations of MZVs, and this is the last one I can thought of so far. Based on previous computation, I conjecture that the following hypergeometric ...
2
votes
1answer
34 views

Finding a closed form to a minimum of a function

It's a try to find a closed form to the minimum of the function : Let $0<x<1$ then define : $$g(x)=x^{2(1-x)}+(1-x)^{2x}$$ Denotes $x_0$ the abscissa of the minimum . Miraculously using Slater's ...
2
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2answers
69 views

Converting Circular formulae to Independent functions

I have a set of equations which I'm trying to transform from a recursive relationship to a more absolute/relative notation. Ideally this is to transform row-based logic into a set-based one for SQL. I ...
1
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0answers
19 views

Product of function in closed form

I am looking for class of functions $\cal F$ s.t., for a function $f(x)\in \cal F$ we can express $$\prod\limits_{k<j}(1-f(n)/f(k))$$ in closed form. Few things: $n,k,j\in \mathbb{Z}$ The function ...
4
votes
2answers
182 views

Evaluate $\int_0^1 \log (1-x)\ _3F_2\left(1,1,1;\frac{3}{2},\frac{3}{2};x\right) \, dx$

I encountered a hypergeometric integral while investigating some harmonic sums, that is $$\int_0^1 \log (1-x)\ _3F_2\left(1,1,1;\frac{3}{2},\frac{3}{2};x\right) \, dx$$ Numerically I suspect that it ...
2
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0answers
57 views

Evaluating $\int_0^1\frac{\ln(1+x^2)\text{Li}_2(x)}{x}dx$ without using $\sum_{n=1}^\infty\frac{H_n}{n^3}x^n$

I am trying to evaluate $$I=\int_0^1\frac{\ln(1+x^2)\text{Li}_2(x)}{x}dx$$ Integration by parts yields $$I=\frac58\zeta(4)-\frac12\int_0^1\frac{\ln(1-x)\text{Li}_2(-x^2)}{x}dx$$ Another related ...
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0answers
16 views

What is the closed form of the generating function of the solution to a linear recurrence?

I've seen many special cases of this, so I thought it would be good to provide a more general version. If this has been done before, that's OK. If $f(x) =\sum_{n=0}^{\infty} a_nx^n $ where, for $n \ge ...
1
vote
1answer
59 views

Closed form of an integral that seems “alike” the Laplace integral

It is well-known that the Laplace integrals $$ \int_0^{+\infty} \frac {\cos (ax)}{b^2 + x^2} \mathrm dx, \quad \int_0^{+\infty} \frac {x\sin (ax)}{b^2 + x^2} \mathrm dx $$ are computable to get a ...
0
votes
0answers
23 views

Closed form of sum of number of digits from 1 to k

From this question The number of digits from $1$ to $k$ is: $$ \text{digits(k)} = n(k + 1) - \dfrac{10^n - 1}{9} $$ where $n$ is the number of digits of $k$ (That is, $n = \lfloor log_{10}(k) \rfloor +...
3
votes
2answers
45 views

Closed form expression for sequence of values created by differently signed series

Consider a sequence of terms of powers of $m\in\mathbb{R}$ as $$ M_n = m^0, m^1, m^2, m^3, \ldots, m^n $$ and create a set that contains all the values of the various signed combinations of these ...
0
votes
1answer
17 views

Find the global max and min of $f(x,y)=18-6x+10y$ on a closed triangular region with vertices $(0,0),(10,0),(10,11)$

So I found $f_x=-6$ and $f_y=10$, which then $-6=0$ and $10=0$. This is were I'm confused, because I'm not sure what my critical points are. and When its along the $x=0$ it is should be $0\leq y \leq ...
1
vote
2answers
64 views

Can we find a closed form for $\sum _{i=0}^{\infty } ((-1)^{i}x^i\prod_{j=1}^{i}\frac{e}{e^j-1})$?

Can we find a closed form for this infinite sum / product? \begin{align*} f(x) &= \sum _{i=0}^{\infty } \biggl((-1)^{i}x^i\prod_{j=1}^{i}\frac{e}{-1 + e^j} \biggr) \\ &= 1 - \biggl(\frac{e}{-1 ...
0
votes
1answer
42 views

Closed form of $a_{k+2} = 4a_k + c^2$ when $c$ is some constant.

What is the closed form of $a_{k+2} = 4a_k + c^2$ when $c$ is some constant. How can we find the closed form of this recurrence with constant? Usually , I’ll use the characteristic root technique, but ...
0
votes
0answers
8 views

Generalization of Hermite and Leguerre polynomials. Terminology and closed form expression.

Is there name for the class of polynomials of the following general form? $P_{k,l,m}^{a}(x)=e^{-ax^{k}}\frac{d^{l}}{dx^{l}}(x^{m}e^{ax^{k}})$ The Hermite polynomials can be expressed as following: ${...
0
votes
0answers
32 views

Closed form solution of systems defined on time-scales (time scale calculus unifies discrete and continuous systems theory)?

Considering the solution of the differential equation \begin{equation} \dot{\rho}=-2L\rho-\gamma(\rho^{2}+1),\,\,\rho(0)=\theta^{-1} \end{equation} where $\theta \in (0,1)$. The time taken for the ...
0
votes
0answers
18 views

Real and imaginary part of elliptic functions

Is there a closed form expression for the real and imaginary parts of elliptic functions like $\operatorname{sn}(u+iv,k)$ and $\operatorname{cn}(u+iv,k)$? Because we can easily separate $\sin(x+iy)$ ...
5
votes
2answers
145 views

Quadratic Euler sums $\sum _{n=1}^{\infty } \frac{(-1)^{n-1} \widetilde H(n)^3}{2 n+1}$

I am investigating a class of quadratic sums of weight 4 and there're only 3 sums remain. Denote $H(n)=\sum_1^n \frac{1}{k}, \widetilde H(n)=\sum_1^n \frac{(-1)^{k-1}}{k}$ the non-alternating/ ...
0
votes
0answers
25 views

Closed formula for sum of quadratic and $1$-norm

Consider a sum of a quadratic and the $1$-norm: $$ f(x) = x^tQx + a^tx + b + ||x||_1,$$ with $Q$ spd, $b \in \mathbb{R}, a \in \mathbb{R}^n$. Is there a closed formula for computing the minimizer of ...
2
votes
1answer
81 views

How to compute the integral $\int_0^1 \frac{x\ln x}{\ln (1-x)}dx$? [duplicate]

How to compute the integral $\int_0^1 \frac{x\ln x}{\ln (1-x)}dx$? We can write \begin{align*} \int_0^1 \frac{x\ln x}{\ln (1-x)}dx=\int_0^1\frac{(1-u)\ln (1-u)}{\ln u}du =-\int_0^1\frac{1-u}{\ln u}\...
0
votes
1answer
17 views

Closed form for the number of $\mathbf{q} \in \Bbb{Z}^n$ with $\|\mathbf{q}\|_{\infty} = h$

A problem I am working on involves the number, $A_h^n$, of points $\mathbf{q} = (q_1, \ldots, q_n)$ in the integer lattice $\Bbb{Z}^n \subseteq \Bbb{R}^n$ such that $\|\mathbf{q}\|_{\infty} = \max_{i ...
0
votes
0answers
15 views

Find the exact decay from a discrete sum of powers

In a discrete function of the form $h_n=\sum_k{A_k\,p_k^n}$, with $h_n$ being an unknown vector of real values, $A_k$, $p_k$ vectors of complex numbers, and $n$ integer, is it possible to find out the ...
1
vote
2answers
47 views

Is there a closed form to this summation? [closed]

I cannot get a closed form for $\sum\limits_{r=0}^{m} \frac{(m+r) !}{(m-r)! (2 r)!}$ ‘ Does anyone have any idea on what it is?
4
votes
0answers
96 views

Is the generalization $\sum_{n=1}^\infty\frac{H_{\frac np}}{n^q}$ known in the literature?

I managed to derive the following generalization $$\sum_{n=1}^\infty\frac{H_{\frac np}}{n^q}=(-1)^qp \sum_{n=1}^\infty\frac{H_{pn}}{(pn)^q}-\sum_{j=1}^{q-2}(-p)^{-j}\zeta(q-j)\zeta(j+1)\tag1$$ and ...
-2
votes
0answers
24 views

N-th derivative of $f(g(x))$

It is known that $$\frac{d^n\Bigl(f(x)+g(x)\Bigr)}{dx^n}=\frac{d^nf(x)}{dx^n}+\frac{d^ng(x)}{dx^n}$$ It can also be shown that $$\frac{d^n\Bigl(f(x)g(x)\Bigr)}{dx^n}=\sum_{k=0}^n\binom{n}{k}\frac{d^kf(...
4
votes
3answers
271 views

Closed form of $\int_0^\infty \arctan^2 \left (\frac{2x}{1 + x^2} \right ) \, dx$

Can a closed form solution for the following integral be found: $$\int_0^\infty \arctan^2 \left (\frac{2x}{1 + x^2} \right ) \, dx.$$ I have tried all the standard tricks such as integration by ...
2
votes
3answers
79 views

A closed form for the dilogarithm integral $\int _{ 0 }^{ 1 }{ \frac { \operatorname{Li}_2\left( 2x\left( 1-x \right) \right) }{ x } dx } $

$$\int _{ 0 }^{ 1 }{ \frac { \operatorname{Li}_2\left( 2x\left( 1-x \right) \right) }{ x } dx } $$ when I was solving an infinite series by using the beta function I encountered the above ...
3
votes
1answer
78 views

A parameterized log-sine integral equating to square of arctan

I have a really convoluted proof of the following: $$ (1) \quad \quad - \int_0^\pi \frac{\sin(2 y) \log(\sin (y/2)) } {r + 1/r + 2 \cos{(2 y)} } dy = \big(\arctan(\sqrt{r})\big)^2 $$ I proved it for ...
1
vote
0answers
26 views

How many degrees of freedom does a symplectic form have?

A symplectic 2-form on a $2n$ dimensional manifold has to be closed, nondegenerate and antisymmetric. My question is: Do these conditions imply how many degrees of freedom the symplectic form has? I ...
0
votes
0answers
47 views

Is there a known closed form for these polynomials?

The polynomials are like: $P_0 = 1\\ P_1 = -2+4x\\P_2 = 8-32x+16x^2\\P_3=-48 + 288 x - 288 x^2 + 64 x^3$ ? I have checked Legendre, Chebyshev, Hermite... Any help will be appreciated! This comes ...
1
vote
1answer
41 views

Request for reference links related to $\sum_{k=1}^{\infty}(-1)^k(\zeta(ak) - 1)$

I was curious if there were any papers, Wikipedia articles, or YouTube videos that cover videos similar to the sum in the title. I’ve been looking for a while now, and the closest thing I got to was ...
2
votes
2answers
96 views

Closed form solution of Integral with many “free variables”

I have a pretty complicated expression that I'm interested in integrating. There's a lot of parameters, so it looks pretty involved: $$ \int_{-\infty}^{\infty}d\Delta\frac{W \sqrt{\frac{\log (2)}{\pi ...
2
votes
1answer
61 views

On a log-gamma definite integral

A very famous log-gamma integral due to Raabe is $$\int_0^1 \log \Gamma (x) \, dx = \frac{1}{2} \log (2\pi).$$ Several proofs of this result can be found here. I would like to known about the ...

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