Question

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

score 313 · Answer 1 · 2020-06-30 08:05:32

313

Views

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

Published on 30 Jun 2020 - 8:05

score 138 · Answer 2 · 2026-03-25 11:14:47

138

Views

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

Published on 25 Mar 2026 - 11:14

score 288 · Answer 3 · 2026-03-29 20:27:43

288

Views

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

Published on 29 Mar 2026 - 20:27

score 66 · Answer 4 · 2026-03-30 16:48:37

66

Views

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

Published on 30 Mar 2026 - 16:48

score 177 · Answer 5 · 2026-03-29 04:34:13

177

Views

Recursive to explicit form involving Fibonacci

Published on 29 Mar 2026 - 4:34

score 119 · Answer 6 · 2026-03-26 04:35:12

119

Views

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

Published on 26 Mar 2026 - 4:35

score 378 · Answer 7 · 2026-03-28 14:44:07

378

Views

Prove $\, _6F_5\left(\{\frac12\}_5,\frac{5}{4};\frac{1}{4},\{1\}_4;-1\right)=\frac{2}{\Gamma \left(\frac{3}{4}\right)^4}$ and another

Published on 28 Mar 2026 - 14:44

score 140 · Answer 8 · 2020-07-06 10:57:17

140

Views

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

Published on 06 Jul 2020 - 10:57

score 183 · Answer 9 · 2020-07-06 21:03:00

183

Views

Beautiful closed form of $\sum_{k=0}^{\infty}\frac{(-1)^{k+1}}{2(k+1)^2}\int_0^1\ln (k+1-k^x)+2\ln (1+k^x)dx$

Published on 06 Jul 2020 - 21:03

score 452 · Answer 10 · 2020-07-07 23:12:59

452

Views

Closed Form Solution to System of Equations

Published on 07 Jul 2020 - 23:12

score 319 · Answer 11 · 2020-07-07 23:23:08

319

Views

Closed-Form solution for nested integrals of this polynomial?

Published on 07 Jul 2020 - 23:23

score 100 · Answer 12 · 2020-07-08 18:29:55

100

Views

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

Published on 08 Jul 2020 - 18:29

score 242 · Answer 13 · 2020-07-10 01:49:56

242

Views

Finding the closed form of $\int _0^{\infty }\frac{\ln \left(1+ax\right)}{1+x^2}\:\mathrm{d}x$

Published on 10 Jul 2020 - 1:49

score 69 · Answer 14 · 2026-03-25 16:03:55

69

Views

A conjecture about the solvability of rational equations of transcendental functions by elementary numbers

Published on 25 Mar 2026 - 16:03

score 126 · Answer 15 · 2020-07-10 21:22:24

126

Views

Evaluate in closed form: $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1-x^{2^{n+1}}}$

Published on 10 Jul 2020 - 21:22

List Question

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

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

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

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

Recursive to explicit form involving Fibonacci

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

Prove $\, _6F_5\left(\{\frac12\}_5,\frac{5}{4};\frac{1}{4},\{1\}_4;-1\right)=\frac{2}{\Gamma \left(\frac{3}{4}\right)^4}$ and another

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

Beautiful closed form of $\sum_{k=0}^{\infty}\frac{(-1)^{k+1}}{2(k+1)^2}\int_0^1\ln (k+1-k^x)+2\ln (1+k^x)dx$

Closed Form Solution to System of Equations

Closed-Form solution for nested integrals of this polynomial?

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

Finding the closed form of $\int _0^{\infty }\frac{\ln \left(1+ax\right)}{1+x^2}\:\mathrm{d}x$

A conjecture about the solvability of rational equations of transcendental functions by elementary numbers

Evaluate in closed form: $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1-x^{2^{n+1}}}$

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