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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When is a multiple sum given in closed form?

Let $d$ be a positive integer and $a>0$. Consider a following multiple sum: \begin{equation} {\mathcal S}^{(d)}_a(j) := \sum\limits_{0 \le j_1 \le j_2 \le \dots \le j_d \le j} \prod\limits_{l=1}^d ...
2
votes
1answer
36 views

Identifying closed form of two-parameter sequence

I've come across a two-parameter sequence $a_{nk}$ with $n=1,2,\ldots$ and $k=1,2,\ldots,n$, and I would like to identify a closed expression for it. So far I have the elements $a_{1k}=\{1\}\\ ...
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1answer
41 views

How to solve the recurrence relation $T(n)=aT(n-1)+bn^c$ with $T(1)=1$

How to solve this recurrence relation? $ T(n)=aT(n-1)+bn^c \\T(1)=1,$ where a, b, c are constant. I want to solve it using generating function, but get stuck. Could anybody help me?
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1answer
54 views

Solution of integral with exponential and trigonometric function

I have an integral, and it looks simple enough for me to believe it could have an analytic solution; however I am unable to find it. I was trying "Gradshteyn and Ryzhik's Table of Integrals, Series, ...
2
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0answers
28 views

Arithmetic-quadratic mean and other “means by limits of means”

For $x,y$ positive real numbers, and $p\neq 0$ real, define the Hölder $p$-mean $$M_p(x,y) := \left(\frac{x^p+y^p}{2}\right)^{1/p}$$ whereas $$M_0(x,y) := \sqrt{xy}$$ is the limit of $M_p(x,y)$ when ...
0
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0answers
14 views

Getting closed-form for $f(x) = \sum_{k=0}^n (f(x-c_1 k-c_2)+1)$

I have to get closed-form for the recursive function $f(x) = \sum_{k=0}^n (f(x-c_1 k-c_2)+1)$ Where $c_1,c_2 \in \mathbb{N}$ $f(x) = 0 \,\,$ for $\,\, 0 < x < c_2$ $f(x) = -1\,\,$ for ...
2
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1answer
52 views

Vandermonde's identity and the close form of $\sum_{k=0}^r C(n,k) C(m,r-k) x^k$

I have a question related to Vandermonde's identity: From Vandermonde's identity, we have: $$ \binom{n+m}{r}=\sum_{k=0}^r \binom{n}{k}\binom{m}{r-k} $$ Now, I have an extra term $x^k$ inside the sum, ...
0
votes
2answers
58 views

Closed form of $\int_{\delta_1}^{\delta_2}(1+Ax)^{-L}x^{L}\exp\left(-Bx\right)dx$

Is there a closed-form expression for the following definite integral? \begin{equation} \mathcal{I} = \int_{\delta_1}^{\delta_2}(1+Ax)^{-L}x^{L}\exp\left(-Bx\right)dx, \end{equation} where $A$, $B$, ...
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1answer
141 views

hunting for the closed form of a series

Let $N$ be a positive integer, and $$ F(N) = \sum_{n=1}^{N} \frac{1-\cos\left(\frac{(2n-1)\pi}{2N}\right)} ...
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1answer
68 views

Closed-form of a series relating to trigonometric function

Occasionally, we may meet some huge expressions, see $$ F(a)=\frac{1}{4}\sum_{n=1}^N \frac{\sin^2\big(\frac{(2n-1)\pi}{2N}\big)}{\Big[a^2-2a\cos \big(\frac{(2n-1)\pi}{2N}\big)+1\Big]^2} $$ where $a$ ...
1
vote
0answers
62 views

Potential function for an exact closed differential form and a solution of a corresponding linear first order homogeneous differential equation

A linear differential form $\sum_{i}\mathcal{E}_{i}(q)\, dq_{i}$ is an exact differential if the conditions $\partial\mathcal{E}_{i}(q)/\partial q_{j}=$ $\partial\mathcal{E}_{j}(q)/\partial q_{i}$ are ...
1
vote
1answer
60 views

How to know if I can't solve an equation with “standard” methods?

I'm particularly fascinated by transcendental equations whose posses closed form solutions and when I pose some of them to my friends or teachers I heard a lot of "You can't solve this in closed form" ...
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2answers
47 views

closed-form of an integral with regard to $a$ [closed]

Suppose that $a$ is a constant and $a>1$. So how can we evaluate the integral $$ I(a) = \int_0^1 \frac{t}{(a-t)\sqrt{1-t^2}}\;dt $$ I just wonder if there is a closed-form. Thank you.
3
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2answers
77 views

Closed form or approximation of $\sum\limits_{i=0}^{n-1}\sum\limits_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)} (i + 2x)(j +2x)$

During the solution of my programming problem I ended up with the following double sum: $$\sum_{i=0}^{n-1}\sum_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)}\cdot (i + 2x)(j +2x)$$ where $x$ is some ...
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1answer
75 views

I know the following integral can be computed in closed form, but I can't figure out how …

The following integral comes up for me when I'm computing a normalizing constant for a probability distribution: $$\int_0^\infty ...
15
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1answer
353 views

Conjectured closed form for $\sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n+\frac{1}{\sqrt{2}}}$

I was trying to find closed form generalizations of the following well known hyperbolic secant sum $$ \sum_{n=-\infty}^\infty\frac{1}{\cosh\pi ...
2
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2answers
68 views

Writing sigma notation $\sum^n_{i=1} \frac {i}{2^i}$ in closed form

What would be a way to find the closed form of $\frac {1}{2} + \frac {2}{4}+\frac {3}{8}+\cdots+\frac {n}{2^n}=\sum^n_{i=1} \frac {i}{2^i}=s$ I've looked at $\frac {s}{2}=\frac {1}{4} + \frac ...
4
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0answers
55 views

Non-existence of closed-form solutions

An equation like $$a^x+b^x=1$$ can be turned to the form $$t^\alpha+t=1$$ by a suitable change of variable. When $\alpha$ is a rational we can put that in a polynomial form $$u^p+u^q=1$$ and ...
4
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0answers
139 views

Simplify $\int_0^\infty \frac{\text{d}{x}}{e^x+x^n}$

I seem to have seen quite a lot of integrals in the form: $$\int_0^\infty \frac{\text{d}x}{e^x+(1+x^n)}$$ But none of those hold a closed forms (at least to my knowledge) ...
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1answer
70 views

Closed form of $\int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx$

Is there a closed-form expression for the following definite integral? \begin{equation} \int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx, \end{equation} where $A$, $B$, and $C$ are ...
8
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2answers
178 views

Integral $\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx$

I found this intriguing integral: $$\int_0^\infty\Big[\log\left(1+x^2\right)-\psi\left(1+x^2\right)\Big]dx\approx0.84767315533332877726581...$$ where $\psi(z)=\partial_z\log\Gamma(z)$ is the digamma. ...
2
votes
4answers
106 views

Quick way to get closed form for this recurrence?

Is there supposed to be a fast way to compute recurrences like these? $T(1) = 1$ $T(n) = 2T(n - 1) + n$ The solution is $T(n) = 2^{n+1} - n - 2$. I can solve it with: Generating functions. ...
16
votes
2answers
317 views

Conjecture ${\large\int}_0^\infty\left[\frac1{x^4}-\frac1{2x^3}+\frac1{12\,x^2}-\frac1{\left(e^x-1\right)x^3}\right]dx=\frac{\zeta(3)}{8\pi^2}$

I encountered the following integral and numerical approximations tentatively suggest that it might have a simple closed form: ...
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1answer
24 views

Closed form of this binomial expression?

Does a closed form for this binomial expression exists? $\sum_{K=2}^{N}\binom{N}{K}P^{K}(1-P)^{N-K}$ Thank you.
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2answers
103 views

Closed form of $\sum\frac{1}{k}$ where $k$ has only factors of $2,3$

Consider the set containing $A$ all positive integers with no prime factor larger than $3$, and define $B$ as $$ B= \sum_{k\in A} \frac{1}{k} $$ Thus, the first few terms of the sum are: ...
15
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4answers
353 views

Need help with $\int_0^\infty\arctan\left(e^{-x}\right)\,\arctan\left(e^{-2x}\right)\,dx$

I was able to calculate: $$\int_0^\infty\arctan\left(e^{-x}\right)\,dx=G$$ $$\int_0^\infty\arctan^2\left(e^{-x}\right)\,dx=\frac\pi2\,G-\frac78\zeta(3)$$ $G$ is the Catalan constant. In both cases ...
6
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0answers
250 views

Rational series representation of $e^\pi$

This question is related to Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$? by Tito Piezas III. Andrew Fraker (2014) found an almost-integer which is equivalent to the following ...
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0answers
43 views

How to solve even/odd divide-and-conquer problems?

I am looking into something called the Josephus problem, which seems to be popular, so I am sure there are lots of explanations online, but I want to do the work myself, but I do need a small push to ...
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0answers
43 views

OEIS A249665 generating function

I'm stuck at finding the general term of the sequence $$1, 1, 1, 2, 6, 14, 28, 56, 118, 254, 541, 1140, 2401, 5074, \ldots$$ According to OEIS, Colin Barker conjectured the recurrence relation to be ...
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0answers
61 views

integral involving error function (erf)

Does anybody know if a closed form of this integral exist? $\int \mbox{erf}(x) \ln(\mbox{erf}(x)) \Bbb dx$ where erf is so called error function. In case there is no closed form solution. Is it ...
2
votes
1answer
129 views

Extract imaginary part of $\text{Li}_3\left(\frac{2}{3}-i \frac{2\sqrt{2}}{3}\right)$ in closed form

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. $\mathrm{Re}[\text{Li}_2(i)]=-\frac{\pi^2}{48}$ Is there a closed form (free of polylogs and ...
3
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0answers
111 views

What is asymptotics of this oscillatory double sum? (Fractal Dimension problem)

The term Gibbs Phenomenon refers to the peculiar way Fourier Series behave at sharp changes in a function's value. However, this problem becomes particularly annoying to deal with when trying to ...
5
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2answers
127 views

How to solve this integral $\int _0^{\infty} e^{-x^3+2x^2+1}\,\mathrm{d}x$

My classmate asked me about this integral:$$\int _0^{\infty} e^{-x^3+2x^2+1}\,\mathrm{d}x$$ but I have no idea how to do it. What's the closed form of it? I guess it may be related to the Airy ...
2
votes
2answers
111 views

Integrate the square root of the ratio of two quadratic polynomials

$$\int \sqrt{\frac{x^2+x-1}{x^2-1}} dx$$ I have been trying to find this integral for a while and I just can't. Does it even have a closed form?
2
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2answers
126 views

How to solve $\int\ x^{\ln x} dx$? [closed]

How to solve this integral $$\int\ x^{\ln x} dx$$ step by step?
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3answers
210 views

Conjecture $\sum_{m=1}^\infty\frac{y_{n+1,m}y_{n,k}}{[y_{n+1,m}-y_{n,k}]^3}\overset{?}=\frac{n+1}{8}$, where $y_{n,k}=(\text{BesselJZero[n,k]})^2$

While solving a quantum mechanics problem using perturbation theory I encountered the following sum $$ S_{0,1}=\sum_{m=1}^\infty\frac{y_{1,m}y_{0,1}}{[y_{1,m}-y_{0,1}]^3}, $$ where ...
10
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2answers
536 views

Integral ${\large\int}_0^{\pi/2}\arctan^2\!\left(\frac{\sin x}{\sqrt3+\cos x}\right)dx$

I need to evaluate this integral: $$I=\int_0^{\pi/2}\arctan^2\!\left(\frac{\sin x}{\sqrt3+\cos x}\right)dx$$ Maple and Mathematica cannot evaluate it in this form. Its numeric value is ...
12
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2answers
292 views

Is there a closed form expression for the “generalized” addition of the first $n$ numbers?

Firstly, I will explain what I am trying to do intuitively. We take the sum of the first $n$ positive integers. Let's say this sum is equal to $q$. Then you add that sum to the sum of the first $q$ ...
10
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1answer
199 views

Need help with $\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$

Please help me to evaluate this integral: $$\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$$ Using substitution $x=2\arctan t$ it can be transformed to: ...
4
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1answer
73 views

Is there a closed form for $n^k$ in terms of $\Delta n^{k+1},\Delta n^k$, …?

Let $\Delta$ be a sort of difference operator on a function $f(n)$ such that $$\Delta f(n)=f(n+1)-f(n)$$ Take the basic power function $f(n)=n^k$, $k\in\mathbb{N}\cup\{0\}$. Then we get ...
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1answer
28 views

What is the closed form of this series: $\sum_{n\geq 1}\frac{n^k{(-1)}^{n+1}}{n!}$ for $k<-10$ and for $k>1$?

I would like to check the closed form of this sum $$\sum_{n\geq 1}\frac{n^k{(-1)}^{n+1}}{n!}$$ , for an integer $k>1$ and $k<-10$. Note : I run some computation in wolfram alpha i have got ...
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1answer
75 views

Closed form for $\sum_{k=1}^n\frac{1}{(2k-1)(2k+1)}$ [closed]

Find the closed form of $$\sum_{k=1}^n\frac{1}{(2k-1)(2k+1)}$$
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1answer
64 views

What is the equation of $F(x)$, given the outputs?

I can't figure out the equation of $F(x)$, if any, for the following sequence of numbers. $$225, 232, 244, 262, 287, 318, 354, 397, 446, 502, 563, 630, 704, 784, 870, 962$$ The equation should ...
4
votes
1answer
78 views

Calculating the value of $\lfloor(1+\sqrt{2})^{2n}\rfloor$

Problem: Calculate the value of $\lfloor(1+\sqrt{2})^{2n}\rfloor$ where $n$ is an arbitrary non-negative integer and $\lfloor x\rfloor$ indicates the largest integer not greater than $x$. What I ...
4
votes
2answers
86 views

closed form of $\sum_{k=0}^n {2n\choose 2k}2^k$

Is it possible to find a closed form for the expression below? $$\sum_{k=0}^n {2n\choose 2k}2^k$$ I have tried counting in two ways but made no progress. And I don't know any combinatorial ...
3
votes
1answer
75 views

Closed form of a generating function $\sum _{n=1}^\infty x^{n^2}$

I am looking for a closed form of the expression $$F(x) = \sum _{n=1}^\infty x^{n^2} $$ The question arose when I attempted to prove Lagrange's four square theorem via generating functions. It ...
10
votes
1answer
246 views

Closed form to an interesting series: $\sum_{n=1}^\infty \frac{1}{1+n^3}$

Intutitively, I feel that there is a closed form to $$\sum_{n=1}^\infty \frac{1}{1+n^3}$$ I don't know why but this sum has really proved difficult. Attempted manipulating a Mellin Transform on the ...
3
votes
1answer
67 views

Is there a closed form for the product of odd zetas?

$$\prod_{n=1}^\infty \zeta(2n+1)=\zeta(3)\zeta(5)\cdots$$ I have only managed to prove that this converges due to comparison with Euler's formula for $\zeta(2n)$ Is there a closed form for that ...
0
votes
1answer
32 views

Closed formula for sum of increasing exponents

I have a sum of the form c¹+c²+...cⁿ. Is it possible to obtain a closed formula for this, and if so how?
5
votes
2answers
103 views

Can one find a closed form solution to $\ln x=\frac{1}{x}$,

Is there a closed form solution of the equation $\ln x=\frac{1}{x}$? I couldn't find a proof myself and I don't know any theorems that says when a closed form solution exists.