Question

Find the closed form of $u_{n+1}=a_nu_n+b_n$

score 205 · Answer 1 · 2026-03-22 11:17:13

205

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Find the closed form of $u_{n+1}=a_nu_n+b_n$

Published on 22 Mar 2026 - 11:17

score 53 · Answer 2 · 2026-03-22 12:50:49

53

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Discrete convolution of generalized pentagonal number sequence

Published on 22 Mar 2026 - 12:50

score 71 · Answer 3 · 2018-02-02 13:57:43

71

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Integral or Closed form expression for $\int^\infty_r \left( 1-\frac{1}{1 + Av^{-\alpha}} \right)v \, dv$

Published on 02 Feb 2018 - 13:57

score 122 · Answer 4 · 2018-02-04 11:53:27

122

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Closed form of $u_n$ with $u_{0} =1$ and $u_n=\frac{1}{3}u_{n-1}-1$?

Published on 04 Feb 2018 - 11:53

score 137 · Answer 5 · 2018-02-05 18:08:17

137

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Computing the product $A_1 A_2 \cdots A_n$ with $A_n =\left(\displaystyle\begin{smallmatrix} n^2&1\\-1&n^2\end{smallmatrix}\right)$

Published on 05 Feb 2018 - 18:08

score 5.4k · Answer 6 · 2026-03-22 20:39:40

5.4k

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Closed-form analytical solutions to Optimal Transport/Wasserstein distance

Published on 22 Mar 2026 - 20:39

score 343 · Answer 7 · 2026-02-28 14:28:27

343

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What is the closed-form of $\sum_{n=1}^\infty\lambda(n)\log\cosh\frac{1}{n}$, where $\lambda(n)$ is the Liouville function?

Published on 28 Feb 2026 - 14:28

score 136 · Answer 8 · 2026-03-22 15:54:52

136

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Simple formula for $H_n = m + \alpha $?

Published on 22 Mar 2026 - 15:54

score 164 · Answer 9 · 2018-02-09 17:05:36

164

Views

How to prove that $ \int_{0}^{\infty} x^n e^{-xt} \sin x\frac{dx}{x} =\frac{i(n-1)!}{2(1+t^2)^n}\left((t-i)^n-(t+i)^n\right)$

Published on 09 Feb 2018 - 17:05

score 436 · Answer 10 · 2026-03-17 22:30:02

436

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Closed form of $I(a)=\int_{0}^a {(e^{-x²})}^{\operatorname{erf}(x)}dx $ and is it behave similar with error function?

Published on 17 Mar 2026 - 22:30

score 2.1k · Answer 11 · 2026-03-19 21:35:15

2.1k

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Example equation which does not have a closed-form solution

Published on 19 Mar 2026 - 21:35

score 59 · Answer 12 · 2018-02-11 09:46:00

59

Views

Upper bound on choosing at least $k$ items out of $n$

Published on 11 Feb 2018 - 9:46

score 100 · Answer 13 · 2018-02-11 12:33:44

100

Views

Deriving the closed form of $M_{n+1} =\frac{2^{2n+1}}{M_n}\left(\sqrt{1+ 2^{-2n}M^2_{n}}-1\right)$

Published on 11 Feb 2018 - 12:33

score 125 · Answer 14 · 2026-03-24 06:19:19

125

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Does a monotone function exist such that there is a "simple" closed form for itself as well as its inverse?

Published on 24 Mar 2026 - 6:19

score 87 · Answer 15 · 2018-02-14 12:56:59

87

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Integrate over sinusoidal region $\int\int(1 - \cos 2\theta ) (1 + \cos(\frac{\theta + 2 \pi u}2 )) du \,d\theta$

Published on 14 Feb 2018 - 12:56

List Question

Find the closed form of $u_{n+1}=a_nu_n+b_n$

Discrete convolution of generalized pentagonal number sequence

Integral or Closed form expression for $\int^\infty_r \left( 1-\frac{1}{1 + Av^{-\alpha}} \right)v \, dv$

Closed form of $u_n$ with $u_{0} =1$ and $u_n=\frac{1}{3}u_{n-1}-1$?

Computing the product $A_1 A_2 \cdots A_n$ with $A_n =\left(\displaystyle\begin{smallmatrix} n^2&1\\-1&n^2\end{smallmatrix}\right)$

Closed-form analytical solutions to Optimal Transport/Wasserstein distance

What is the closed-form of $\sum_{n=1}^\infty\lambda(n)\log\cosh\frac{1}{n}$, where $\lambda(n)$ is the Liouville function?

Simple formula for $H_n = m + \alpha $?

How to prove that $ \int_{0}^{\infty} x^n e^{-xt} \sin x\frac{dx}{x} =\frac{i(n-1)!}{2(1+t^2)^n}\left((t-i)^n-(t+i)^n\right)$

Closed form of $I(a)=\int_{0}^a {(e^{-x²})}^{\operatorname{erf}(x)}dx $ and is it behave similar with error function?

Example equation which does not have a closed-form solution

Upper bound on choosing at least $k$ items out of $n$

Deriving the closed form of $M_{n+1} =\frac{2^{2n+1}}{M_n}\left(\sqrt{1+ 2^{-2n}M^2_{n}}-1\right)$

Does a monotone function exist such that there is a "simple" closed form for itself as well as its inverse?

Integrate over sinusoidal region $\int\int(1 - \cos 2\theta ) (1 + \cos(\frac{\theta + 2 \pi u}2 )) du \,d\theta$

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