Question

On the cubic counterpart of Ramanujan's $\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots$?

score 307 · Answer 1 · 2026-03-27 23:14:16

307

Views

On the cubic counterpart of Ramanujan's $\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots$?

Published on 27 Mar 2026 - 23:14

score 177 · Answer 2 · 2023-10-23 16:25:20

177

Views

Generalising $I_n=\int_0^1 \arcsin(\sqrt{1-x^n})\mathrm dx$.

Published on 23 Oct 2023 - 16:25

score 173 · Answer 3 · 2026-03-26 17:32:46

173

Views

Proving $\sum_{n=-\infty}^\infty n^2e^{-\pi n^2}=\frac{\Gamma (1/4)}{4\sqrt{2}\pi^{7/4}}$

Published on 26 Mar 2026 - 17:32

score 52 · Answer 4 · 2026-03-27 23:14:24

52

Views

Does the rest of this family of continued fractions have closed forms?

Published on 27 Mar 2026 - 23:14

score 186 · Answer 5 · 2026-03-29 03:51:13

186

Views

The closed-form of $1-5\left(\frac{1}{2}\right)^k+9\left(\frac{1\cdot3}{2\cdot4}\right)^k-13\left(\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\right)^k+\dots$?

Published on 29 Mar 2026 - 3:51

score 71 · Answer 6 · 2026-04-03 09:59:09

71

Views

Does this series representation for $\zeta(s)$ converge in the critical strip?

Published on 03 Apr 2026 - 9:59

score 90 · Answer 7 · 2023-10-28 01:51:55

90

Views

Integral relating gamma function's linear subspaces

Published on 28 Oct 2023 - 1:51

score 103 · Answer 8 · 2023-10-29 13:14:24

103

Views

$ e^{-\mu } \sum_{k=m }^{\infty} \frac{\mu^k}{k!} \le (\frac{\mu}{m})^m e^{m-\mu} $ when $1\le \mu \le m$

Published on 29 Oct 2023 - 13:14

score 194 · Answer 9 · 2026-03-27 07:49:27

194

Views

Is Beta function essential to evaluating the integral $\int_0^{\infty} \frac{d x}{\left(1+x^4\right)^n}$?

Published on 27 Mar 2026 - 7:49

score 60 · Answer 10 · 2026-03-26 06:26:42

60

Views

pochammer symbol and hypergeometric function

Published on 26 Mar 2026 - 6:26

score 316 · Answer 11 · 2026-03-26 12:36:33

316

Views

Values for which $\frac{K(k')}{K(k)}=a+i$ where $a$ is an algebraic number.

Published on 26 Mar 2026 - 12:36

score 31 · Answer 12 · 2026-03-30 13:21:35

31

Views

Covariance of $\frac{X}{\|X\|}$ with $X\sim N(\mu, Id)$, and ${}_1F_1(a;b;z)$ properties

Published on 30 Mar 2026 - 13:21

score 42 · Answer 13 · 2026-04-02 06:44:57

42

Views

Limit of sum of ratios of gamma function like products

Published on 02 Apr 2026 - 6:44

score 84 · Answer 14 · 2026-04-02 06:42:33

84

Views

Question about derivative of the falling factorial

Published on 02 Apr 2026 - 6:42

score 48 · Answer 15 · 2023-11-04 08:03:31

48

Views

Could someone help me with this improper integral?

Published on 04 Nov 2023 - 8:03

List Question

On the cubic counterpart of Ramanujan's $\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots$?

Generalising $I_n=\int_0^1 \arcsin(\sqrt{1-x^n})\mathrm dx$.

Proving $\sum_{n=-\infty}^\infty n^2e^{-\pi n^2}=\frac{\Gamma (1/4)}{4\sqrt{2}\pi^{7/4}}$

Does the rest of this family of continued fractions have closed forms?

The closed-form of $1-5\left(\frac{1}{2}\right)^k+9\left(\frac{1\cdot3}{2\cdot4}\right)^k-13\left(\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\right)^k+\dots$?

Does this series representation for $\zeta(s)$ converge in the critical strip?

Integral relating gamma function's linear subspaces

$ e^{-\mu } \sum_{k=m }^{\infty} \frac{\mu^k}{k!} \le (\frac{\mu}{m})^m e^{m-\mu} $ when $1\le \mu \le m$

Is Beta function essential to evaluating the integral $\int_0^{\infty} \frac{d x}{\left(1+x^4\right)^n}$?

pochammer symbol and hypergeometric function

Values for which $\frac{K(k')}{K(k)}=a+i$ where $a$ is an algebraic number.

Covariance of $\frac{X}{\|X\|}$ with $X\sim N(\mu, Id)$, and ${}_1F_1(a;b;z)$ properties

Limit of sum of ratios of gamma function like products

Question about derivative of the falling factorial

Could someone help me with this improper integral?

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