Question

What is the general formula of the sum $\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{m}$ for $m,n\in\mathbb{N}$?

score 506 · Answer 1 · 2026-02-23 13:05:01

506

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What is the general formula of the sum $\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{m}$ for $m,n\in\mathbb{N}$?

Published on 23 Feb 2026 - 13:05

score 119 · Answer 2 · 2021-09-20 02:39:34

119

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Ask for a proof of logarithmically complete monotonicity of a power-exponential function involving the difference of the psi and logarithmic functions

Published on 20 Sep 2021 - 2:39

score 61 · Answer 3 · 2021-10-24 21:49:43

61

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Polygamma sum problem

Published on 24 Oct 2021 - 21:49

score 63 · Answer 4 · 2021-10-26 16:49:34

63

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Polygamma sum problem ...

Published on 26 Oct 2021 - 16:49

score 218 · Answer 5 · 2021-10-28 19:02:47

218

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New trigamma identity for $\Psi_1(\frac3{20})+6\,\Psi_1(\frac15)+10\,\Psi_1(\frac25)-\Psi_1(\frac1{20})$

Published on 28 Oct 2021 - 19:02

score 366 · Answer 6 · 2021-11-02 18:04:11

366

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In an attempt to find $I = \int_0^\infty \frac{t}{e^t-1}dt$

Published on 02 Nov 2021 - 18:04

score 70 · Answer 7 · 2021-11-29 20:48:04

70

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Simplification of a difficult identity involving the digamma function

Published on 29 Nov 2021 - 20:48

score 214 · Answer 8 · 2021-12-07 20:43:14

214

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Great difficulty in finding the residues of $\frac{\mathrm{Log}\Gamma\left(\frac{z+ai}{2\pi i}\right)}{\cosh(z)+1}$

Published on 07 Dec 2021 - 20:43

score 110 · Answer 9 · 2021-12-18 14:02:50

110

Views

Evaluate the limit with the Appell F1 function

Published on 18 Dec 2021 - 14:02

score 120 · Answer 10 · 2026-03-25 12:20:35

120

Views

Sum of Fermi-Dirac integrals with opposite chemical potentials: closed form (Le Bellac eq. 1.13)

Published on 25 Mar 2026 - 12:20

score 125 · Answer 11 · 2022-04-05 09:53:42

125

Views

Closed form for $\rm{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)$

Published on 05 Apr 2022 - 9:53

score 85 · Answer 12 · 2022-05-24 17:38:50

85

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Computing solutions to polygamma function of order 1?

Published on 24 May 2022 - 17:38

score 57 · Answer 13 · 2022-08-01 16:38:23

57

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Can somebody show me how to derive the identity relating the polygamma function and the lerch transcendent

Published on 01 Aug 2022 - 16:38

List Question

What is the general formula of the sum $\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}\binom{k/2}{m}$ for $m,n\in\mathbb{N}$?

Ask for a proof of logarithmically complete monotonicity of a power-exponential function involving the difference of the psi and logarithmic functions

Polygamma sum problem

Polygamma sum problem ...

New trigamma identity for $\Psi_1(\frac3{20})+6\,\Psi_1(\frac15)+10\,\Psi_1(\frac25)-\Psi_1(\frac1{20})$

In an attempt to find $I = \int_0^\infty \frac{t}{e^t-1}dt$

Simplification of a difficult identity involving the digamma function

Great difficulty in finding the residues of $\frac{\mathrm{Log}\Gamma\left(\frac{z+ai}{2\pi i}\right)}{\cosh(z)+1}$

Evaluate the limit with the Appell F1 function

Sum of Fermi-Dirac integrals with opposite chemical potentials: closed form (Le Bellac eq. 1.13)

Closed form for $\rm{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)$

Computing solutions to polygamma function of order 1?

Can somebody show me how to derive the identity relating the polygamma function and the lerch transcendent

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