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 507 · Answer 1 · 2026-04-09 16:18:31

507

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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 09 Apr 2026 - 16:18

score 120 · Answer 2 · 2026-03-25 20:36:05

120

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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 25 Mar 2026 - 20:36

score 62 · Answer 3 · 2026-03-25 20:40:09

62

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Polygamma sum problem

Published on 25 Mar 2026 - 20:40

score 64 · Answer 4 · 2026-03-25 20:40:08

64

Views

Polygamma sum problem ...

Published on 25 Mar 2026 - 20:40

score 219 · Answer 5 · 2026-03-26 01:12:45

219

Views

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

Published on 26 Mar 2026 - 1:12

score 367 · Answer 6 · 2026-03-25 23:38:14

367

Views

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

Published on 25 Mar 2026 - 23:38

score 71 · Answer 7 · 2026-03-25 20:34:45

71

Views

Simplification of a difficult identity involving the digamma function

Published on 25 Mar 2026 - 20:34

score 215 · Answer 8 · 2026-03-26 01:23:40

215

Views

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

Published on 26 Mar 2026 - 1:23

score 111 · Answer 9 · 2026-03-25 20:41:14

111

Views

Evaluate the limit with the Appell F1 function

Published on 25 Mar 2026 - 20:41

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 126 · Answer 11 · 2026-03-25 20:42:10

126

Views

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

Published on 25 Mar 2026 - 20:42

score 86 · Answer 12 · 2026-03-25 20:41:22

86

Views

Computing solutions to polygamma function of order 1?

Published on 25 Mar 2026 - 20:41

score 58 · Answer 13 · 2026-03-25 22:06:31

58

Views

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

Published on 25 Mar 2026 - 22:06

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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