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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6
votes
1answer
79 views

Summation of a series involving powers of Fibonacci numbers.

I'm interested in this series: $$\mathcal S_p=\sum_{n=1}^\infty\frac{\left(F_n\right)^p}{2^{np}},\quad p\in\mathbb N,\tag1$$ where $F_n$ are the Fibonacci numbers, defined by the recurrence ...
3
votes
1answer
136 views

Prove that $\int_{-\infty}^\infty \frac{\operatorname{Ai}^2(x+a_n)}{x^2}dx = 1$

While I've been thinking about this question, I've found that for all $n \geq 1$ integer values, we have $$ \mathcal{I}_n = \int_{-\infty}^\infty \frac{\operatorname{Ai}^2(x+a_n)}{x^2}dx ...
2
votes
0answers
37 views

Solving the logarithmic rational equation

I'm wondering there exist the way to solve the equation form of: $$ \log f(x) + g(x) = c $$ where $f(x)$ and $g(x)$ are rational functions, $c$ is a constant. Is there any general(in closed form) ...
2
votes
1answer
63 views

Reference to proof of closed form for of $\zeta_{\text{Ai}}(2)$

I was stunned to find out we can evaluate the Airy Zeta Function at $s = 2$ exactly: $$ \zeta_{\text{Ai}}(2) \equiv \sum_{k=1}^\infty a_k^{-2}=\frac{3^{5/3}}{4\pi^2}\Gamma^4\left(\frac23\right) $$ ...
2
votes
2answers
56 views

Closed form of $\exp(\sum_{n = 1}^{\infty} \frac{t^n}{n})$

I want to show that $$\exp(\sum_{n = 1}^{\infty} \frac{t^n}{n}) = \frac{1}{1 - t},$$ for $t \in (-1, 1)$. Using derivatives and fundamental theorem of calculus, I have a basic idea how to proceed. ...
3
votes
0answers
49 views

Any closed form for this expression?$ \sum_{k=0,\,l=0}^{k=n,\,l=m}\frac{\lambda^{l+k}}{k!\,l!}\sqrt{\frac{n!\,m!}{(n-k)!(m-l)!}}\delta_{n-k,\,m-l}$

I am looking for a closed form of this expression. If you have seen something like this or remember something similar, please let me know. My sincere thank! $$ ...
3
votes
1answer
65 views

How to solve a functional differential equation?

$$(1) \quad \cfrac{d}{dx} (f(x^n))=\cfrac{-f(x^n)^2}{f(n \cdot x^{n-1})}$$ How do I solve this functional differential equation? I need a closed form solution, so approximations won't cut it, I'll ...
3
votes
2answers
77 views

Improper integral of $\frac{\sin x}{x}e^{-ax}$

For $a>0$ define $$I(a)=\int_0^\infty \frac{\sin x}{x}e^{-ax} \, dx,$$ I can show it is continuous at $0$, but by differentiating in $a$, I can't see why $$I(a)=\frac{\pi}{2}-\arctan(a).$$ Thanks ...
0
votes
1answer
80 views

Finding a closed form of recursive formula $T(n)=4T(n-1) - 4T(n-2)$

Find the closed form for the following: $$T(n) = \begin{cases} 1\quad &\text{ if } n = 0 \\ 4\quad &\text{ if } n = 1 \\ 4T(n-1) - 4T(n-2) & \text{ if } n > 1 \end{cases}$$ ...
0
votes
0answers
38 views

Looking for rounded corner plane curve with certain properties (SIDESTEPPED)

For a project involving simulating traffic lights, I am currently looking for a formula to get a rounded 90-degree corner (to describe the path of a turning car) with certain properties: Defined in ...
6
votes
1answer
178 views
4
votes
1answer
72 views

Closed form for ${\large\int}_0^1x\,\operatorname{li}\!\left(\frac1x\right)\ln^{1/4}\!\left(\frac1x\right)dx$

Let $\operatorname{li}(x)$ denote the logarithmic integral: $$\operatorname{li}(x)=\int_0^x\frac{dt}{\ln t}.$$ How can we prove the following conjectured closed form? ...
8
votes
3answers
127 views

Is there a closed form for the nontrivial solutions of $x^y = y^x$?

It can be shown $\forall x \in \Bbb{R}^+ \exists \{y_1, y_2\} : x^y = y^x.$, and the first of these two numbers is trivial, $y_1 = x$. The second is nontrivial, and I cannot find a closed form for all ...
3
votes
2answers
81 views

Finding $S=\sum_2^{\infty}\frac{\ln(n+1)}{(n^2-1)}$

Is there any chance to express the series $$S=\sum_2^{\infty}\frac{\ln(n+1)}{(n^2-1)}$$ in terms of a known function? My idea is to start from the generalized-Euler-constant function ...
5
votes
1answer
81 views

Closed form for finite sum of ${\rm csch}^2$

In a recent problem I was attempting to solve, I hit a road block when I reduced the problem to that of finding a closed form for the following sum $$ \mathcal{S}_n(x)\equiv\sum_{k=1}^n{\rm ...
4
votes
4answers
127 views

Compute a double integral: $\int_0^1\int_0^1 \frac{(xy)^s}{\sqrt{-\log(xy)}}dxdy$

Is it possible to get a closed form of the following integral? $$I=\int_0^1\int_0^1 \frac{(xy)^s}{\sqrt{-\log(xy)}} \, dx \, dy\quad \quad \quad (s>0).$$ My attempt: I’ve tried a change of ...
1
vote
1answer
53 views

Can this integral be computed in closed-form?

I am interested in computing a normalizing constant (of a Gaussian density in dimension $3$). Such normalizing constants often do not have a closed form. In dimension $2$, this normalizing constant ...
5
votes
1answer
179 views

Strategies for evaluating sums $\sum_{n=1}^\infty \frac{H_n^{(m)}z^n}{n}$

I'm looking for strategies for evaluating the following sums for given $z$ and $m$: $$ \mathcal{S}_m(z):=\sum_{n=1}^\infty \frac{H_n^{(m)}z^n}{n}, $$ where $H_n^{(m)}$ is the generalized harmonic ...
5
votes
1answer
112 views

Closed form for $\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$

Does anyone know if there happens to exist a closed form solution for this sum: $$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$ For low values of $a$, Wolfram Alpha gives a closed form in terms ...
-1
votes
1answer
43 views

Is it possible to find a closed form for this recursive sequence?

Is it possible to find a closed form for the sequence defined by $$\begin{align*} a_0&=3\\ na_n&=(n-1)a_{n-1}+1\quad\text{for }n\ge 1\;. \end{align*}$$
0
votes
1answer
56 views

Closed-form solution for $x^a = (1-x)^{1-a}\cdot b$ with $0 < a < 1$

How can I tell if $x^a = (1-x)^{1-a} \cdot b$ has a closed form solution for $x$, assuming $0<a<1$. It does in the case of $a=\frac{1}{2}$, but is this the only case? Computer algebra systems ...
7
votes
1answer
112 views

Closed form for $\int_0^R \frac{dx}{\sqrt{\ln(1+x)}}$, R>0

I stumbled on an interesting integral doing some physics exercise which did not require its closed form (if it has any). It has, however, sparked my interest and I tried my best to find it, but I ...
3
votes
1answer
58 views

Is there any closed form of the following nested series?

I am wondering if there is any closed form of the following summation? $\sum \limits_{i=0}^{\infty} (q^i \sum \limits_{j=0}^i \dfrac{a^j}{j!})$ where |q|<1. I know that $\sum \limits_{i=0}^{\infty} ...
9
votes
1answer
155 views

Closed Form for $~\lim\limits_{n\to\infty}~\sqrt n\cdot(-e)^{-n}\cdot\sum\limits_{k=0}^n\frac{(-n)^k}{k!}$

$\qquad\qquad\qquad$ Does $~\displaystyle\lim_{n\to\infty}\frac{\sqrt n}{(-e)^n}\cdot\sum_{k=0}^n\frac{(-n)^k}{k!}~$ possess a closed form expression ? Inspired by this frequently asked question, ...
0
votes
1answer
41 views

Convergence and sum of an infinite series: $\sum_{i=1}^{\infty}\frac{6}{24 i-4 i^2-35}$

Determine whether the following series is convergent or divergent. If convergent, find the sum. $$\sum_{i=1}^{\infty}\frac{6}{24 i-4 i^2-35}$$ Since the limit of the series is zero, I know that it is ...
1
vote
1answer
139 views

Evaluate the sum $\sum_1^n{2k+1\over k(k+1)}$ in closed form

Evaluate the sum $$\sum_{k=1}^n{2k+1\over k(k+1)}$$ in closed form. You are allowed to have the harmonic number $$H_n = \sum_1^nk^{-1}$$ in your closed–form formula. I am having trouble with this ...
8
votes
1answer
95 views

The value of $\sum_{n=0}^{\infty} \, \bigl(\prod_{i=0}^{n-1} q^n-q^i\bigr)^{-1}$

Let $q > 1$. What can we say about the value of $$\sum_{n=0}^{\infty} \, \bigl(\prod\limits_{i=0}^{n-1} q^n-q^i\bigr)^{-1} ~~?$$ The series clearly converges. Is there a closed form or something ...
0
votes
1answer
199 views

Climbing a n-stair staircase, taking 2 or 3 stairs each step…

Suppose a person has a n-stair staircase to climb, and they can go up exactly 2 or 3 stairs each time they take a step. Generate some initial data. Find and explain the recurrence relation to ...
13
votes
3answers
759 views

Why is there no general form for the harmonic numbers?

The Harmonic numbers $H_n$ are given by the sum of the reciprocals of the natural numbers up to a given $n$, ie: $H_1 = 1$ $H_2 = 1 + 1/2 = 3/2$ $H_3 = 1 + 1/2 + 1/3 = 11/6$ $H_n$ for noninteger ...
0
votes
1answer
66 views

Need help finding the closed form of a sequence based upon the fibonacci sequence.

I have been given an assignment question that asks for a simple closed form of the following sequence: $$G_n=\left|\begin{array}{cc} F_n & F_{n+1}\\ F_{n+1} & F_{n+2} \end{array}\right|$$ I ...
5
votes
3answers
193 views

Closed form of: $\displaystyle \int_0^{\pi/2}x^{n}\ln{(\sin x)}dx $

$\displaystyle \int_0^{\pi/2}x^{n}\ln{(\sin x)}dx $ Does a closed form of the above integral exists? $n$ is a positive integer
8
votes
4answers
180 views

Find the solution of $\lfloor{x^2}\rfloor−\lfloor{3x}\rfloor+2=0$

Is anyone able to help me with the following equation concerned the floor function $\lfloor{x^2}\rfloor−\lfloor{3x}\rfloor+2=0$ I don't know how to deal with the floor terms properly.
0
votes
0answers
34 views

Is it possible to find a closed-form solution to this nonlinear system of complex equations?

I have a system of complex equations that looks like this: $(A+j\lambda) \cdot Z=b$, All elements in square matrix $A$ and vector $b$ are known. $\lambda$ is a diagonal matrix of real elements with ...
1
vote
1answer
42 views

closed form for the following integral which is similar to Laplace transform

I want to find a closed form for this integral: $\int\limits_{x=0}^{\infty} \exp(-\frac{1}{x})x^n\exp(-sx)dx$ I know that it has closed form for $n=0$ but what about $n\neq0$? Does anyone have any ...
0
votes
3answers
74 views

Is there a way to write $\dots 2(2(2(2+1)-1)+1)-1 \dots$ in closed form?

It's something like a sequence I'm working with and a part of it has coefficients, which go $3$, $5$, $11$, $21$ and so on. I was wondering if it's possible to find a (closed) formula dependent on say ...
3
votes
2answers
271 views

Finding the general formula of a sequence: $3,8,23,68,203,608,\cdots$

I have the following sequence : $$3,8,23,68,203,608,\cdots$$ I have found that definition by recurrence of this is $$a(n)=3a(n-1)-1$$ where $a_0=3$ as the first term. I want to find the explicit ...
3
votes
3answers
151 views

Prove that $\int_0^\infty\frac{x\cos(x)-\sin(x)}{x\left({e^x}-1\right)}\,dx = \frac{\pi}2+\arg\left(\Gamma(i)\right)-\Re\left(\psi_0(i)\right)$

While I was working on this question, I've found that $$ I=\int_0^\infty\frac{x\cos(x)-\sin(x)}{x\left({e^x}-1\right)}\,dx = \frac{\pi}2+\arg\left(\Gamma(i)\right)-\Re\left(\psi_0(i)\right), $$ where ...
3
votes
1answer
48 views

What is the analytic representation of $\sum_{n=0}^\infty \frac{ a^nx^n}{n!}$?

I only know that a geometric series is $$\frac1{1-x}\qquad (|x|<1)$$ and this looks similar.
6
votes
1answer
124 views

Simplifying a certain polylogarithmic sum in two variables

This question is related to my previous question here. While tinkering around for a solution I found that the integral there can be reduced to the problem of solving the following basic logarithmic ...
7
votes
1answer
336 views

Closed-form of $\int_0^1 \frac{\ln^2(x)}{\sqrt{x(a-bx)}}\,dx$

I'm interesed in the following integral, for $a,b>0$: $$ \mathcal{I}(a,b) := \int_0^1 \frac{\ln^2(x)}{\sqrt{x(a-bx)}}\,dx $$ Mathematica could evaluate it in term of hypergeometric functions, but ...
12
votes
0answers
184 views

Dilogarithm identity containing the tribonacci constant

The motivation of this question is the brilliant conjecture by @Tito Piezas III. In $(4)$ of his question the equation seems to be true for all $n > 1$ real numbers. The case $n=2$ leads us to a ...
1
vote
1answer
228 views

How do I write a closed form expression for $\sum _{i=0}^{n-1}$ in terms of n?

I am given this:$$\sum _{i=1}^n a_i = n^2-n,a_0=4$$ How do I write a closed form expression for $$\sum _{i=0}^{n-1}$$in terms of n? I know that for $$\sum _{i=1}^{n-1}$$ the expression would be ...
0
votes
0answers
42 views

Closed form for a recursive equation that include the ceiling function

Can someone help me with finding the closed form of g(n) in terms of n, A, and B? g(n<0)=0 ; g(0)=0 ; g(1)=0 ; g(n) = A + g(n-1) - ceiling[g(n-1)/B] , n>=2 , A and B are Natural numbers, ...
3
votes
2answers
67 views

Sum of the series $\sum u_n$ where $u_n=\frac{\sqrt{(n-1)!}}{(1+\sqrt{1}) \dots (1+\sqrt{n})}$

While I'm able to prove that the series $u_n=\frac{\sqrt{(n-1)!}}{(1+\sqrt{1}) \dots (1+\sqrt{n})}$ converges, I don't see the trick to compute the value of its sum starting at $n=2$. Any clue on the ...
4
votes
1answer
113 views

What's the $n$-th derivative of $\ln(\sin(x))$?

I want to find the $n$-th derivative of $\ln(\sin x)$, i.e. $$ \frac{d^n\ln(\sin x)}{dx^n} $$ where $x\in (0,\pi/2)$ such that $\sin x>0$. To make the problem definitely, $x=\pi/4$ is assumed. In ...
0
votes
0answers
150 views

Integration with Log of error function (erf)

Can anybody help me evaluating the closed-form or an approximate form of $H(x) = \int P(x) \ln(P(x)) \Bbb dx$ where $P(x) = \frac{C(x)}{v\int C(x) \Bbb dx}$ and $C(x) = {\frac ...
11
votes
2answers
241 views

Conjectured closed form for $\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$

There are few known closed form for values of the dilogarithm at specific points. Sometimes only the real part or only the imaginary part of the value is known, or a relation between several different ...
1
vote
2answers
49 views

Closed form of a power series

Find the function that represents the following sum: $\sum\limits_{k=0} ^\infty \frac{(n^2)}{n!} x^n$. Can't find the function that represents this.
21
votes
2answers
285 views

Integral ${\large\int}_0^1\ln^3\!\left(1+x+x^2\right)dx$

I'm interested in this integral: $$I=\int_0^1\ln^3\!\left(1+x+x^2\right)dx.\tag1$$ Can we prove that ...