Question

Reference request: $q$-gamma, $q$-polygamma, $q$-Pochhammer.

score 85 · Answer 1 · 2020-01-20 01:10:45

85

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Reference request: $q$-gamma, $q$-polygamma, $q$-Pochhammer.

Published on 20 Jan 2020 - 1:10

score 300 · Answer 2 · 2026-03-23 11:13:20

300

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$\int_0^{\pi/2} \sec^a(t)\,dt= \frac{\sqrt{\pi}}{2\Gamma\left(1-\frac{a}{2}\right)}\Gamma\left(\dfrac{1-a}{2}\right)$

Published on 23 Mar 2026 - 11:13

score 304 · Answer 3 · 2020-02-17 02:19:34

304

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Is the closed form for $\sum_{k=1}^\infty\frac{\overline{H}_k}{k^m}$ known in the literature?

Published on 17 Feb 2020 - 2:19

score 51 · Answer 4 · 2020-02-22 04:37:24

51

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Sum involving 2nd antiderivative of the digamma function.

Published on 22 Feb 2020 - 4:37

score 238 · Answer 5 · 2020-02-28 01:17:03

238

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Evaluate $\int _0^{2 \pi }\int _0^{2 \pi }\log (3-\cos (x+y)-\cos (x)-\cos (y))dxdy$

Published on 28 Feb 2020 - 1:17

score 130 · Answer 6 · 2020-03-28 00:23:44

130

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Prove $\lim_{x\to\infty}\sum_{n=1}^x x\log\left(1+\frac1{xn(an+1)}\right)= H_{\frac1a}$

Published on 28 Mar 2020 - 0:23

score 153 · Answer 7 · 2020-04-23 18:37:40

153

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Analytic continuation of $H_x^{(k)}=\sum_{n=1}^x \frac{1}{n^k}$?

Published on 23 Apr 2020 - 18:37

score 66 · Answer 8 · 2020-05-01 15:59:24

66

Views

Decomposition of $\psi^{(n)}(1)$ in terms of $\psi^{(n)}(k)$

Published on 01 May 2020 - 15:59

score 93 · Answer 9 · 2020-05-03 16:35:34

93

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How can we evaluate $\sum_{k\geq 0} \frac{1}{(2k+1)^3}$?

Published on 03 May 2020 - 16:35

score 87 · Answer 10 · 2020-05-06 09:22:29

87

Views

Find the function $f(x)=\sum_{n=1}^{\infty}\frac{H_{n-1}(-x)^n}{n!}$

Published on 06 May 2020 - 9:22

score 159 · Answer 11 · 2020-06-09 19:13:53

159

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How to Prove : $ \gamma +\ln\left(\frac{\pi}{4}\right) = \sum_{n=2}^{\infty} \frac{(-1)^{n} \zeta{(n)}}{2^{n-1}n} $

Published on 09 Jun 2020 - 19:13

score 72 · Answer 12 · 2020-06-19 22:26:12

72

Views

Log gamma integral with a slightly more complicated polynomial factor

Published on 19 Jun 2020 - 22:26

score 181 · Answer 13 · 2020-06-27 23:29:20

181

Views

How can we derive the asymptotic expansion for the second derivative of the gamma-function?

Published on 27 Jun 2020 - 23:29

score 213 · Answer 14 · 2020-07-22 03:11:54

213

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Prove $\lim_{n\mapsto 0}[(\psi(n)+\gamma)\psi^{(1)}(n)-\frac12\psi^{(2)}(n)]=2\zeta(3)$

Published on 22 Jul 2020 - 3:11

score 115 · Answer 15 · 2016-08-17 07:45:20

115

Views

On $\lim_{x\to 0}\frac{-(1+\sqrt{-x})\psi^{(0)}(1-\sqrt{-x})+(1+\sqrt{-x})\psi^{(0)}(\sqrt{-x}+1)+2\psi^{(0)}(1+x)}{2(\sqrt{-x}-1)(\sqrt{-x}+1)x}$

Published on 17 Aug 2016 - 7:45

List Question

Reference request: $q$-gamma, $q$-polygamma, $q$-Pochhammer.

$\int_0^{\pi/2} \sec^a(t)\,dt= \frac{\sqrt{\pi}}{2\Gamma\left(1-\frac{a}{2}\right)}\Gamma\left(\dfrac{1-a}{2}\right)$

Is the closed form for $\sum_{k=1}^\infty\frac{\overline{H}_k}{k^m}$ known in the literature?

Sum involving 2nd antiderivative of the digamma function.

Evaluate $\int _0^{2 \pi }\int _0^{2 \pi }\log (3-\cos (x+y)-\cos (x)-\cos (y))dxdy$

Prove $\lim_{x\to\infty}\sum_{n=1}^x x\log\left(1+\frac1{xn(an+1)}\right)= H_{\frac1a}$

Analytic continuation of $H_x^{(k)}=\sum_{n=1}^x \frac{1}{n^k}$?

Decomposition of $\psi^{(n)}(1)$ in terms of $\psi^{(n)}(k)$

How can we evaluate $\sum_{k\geq 0} \frac{1}{(2k+1)^3}$?

Find the function $f(x)=\sum_{n=1}^{\infty}\frac{H_{n-1}(-x)^n}{n!}$

How to Prove : $ \gamma +\ln\left(\frac{\pi}{4}\right) = \sum_{n=2}^{\infty} \frac{(-1)^{n} \zeta{(n)}}{2^{n-1}n} $

Log gamma integral with a slightly more complicated polynomial factor

How can we derive the asymptotic expansion for the second derivative of the gamma-function?

Prove $\lim_{n\mapsto 0}[(\psi(n)+\gamma)\psi^{(1)}(n)-\frac12\psi^{(2)}(n)]=2\zeta(3)$

On $\lim_{x\to 0}\frac{-(1+\sqrt{-x})\psi^{(0)}(1-\sqrt{-x})+(1+\sqrt{-x})\psi^{(0)}(\sqrt{-x}+1)+2\psi^{(0)}(1+x)}{2(\sqrt{-x}-1)(\sqrt{-x}+1)x}$

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