How is this formula for the Dirichlet $beta$-function derived?
According to Wikipedia, we have:
$${displaystyle beta (2k)={frac {1}{2(2k-1)!}}sum _{m=0}^{infty }left(left(sum _{l=0}^{k-1}{binom {2k-1}{2l}}{frac {(-1)^{l}A_{2k-2l-1}}{2l+2m+1}}right)-{frac {(-1)^{k-1}}{2m+2k}}right){frac {A_{2m}}{(2m)!}}{left({frac {pi }{2}}right)}^{2m+2k},}$$
where ${displaystyle A_{k}}$ is the Euler zigzag number.
However, the citation is missing on Wikipedia. Is there an easy way to derive this, or alternatively, a link to a paper with a proof?
reference-request special-functions beta-function
asked Nov 12 '18 at 13:25
KlangenKlangen
1,65711334
add a comment |
According to Wikipedia, we have:
$${displaystyle beta (2k)={frac {1}{2(2k-1)!}}sum _{m=0}^{infty }left(left(sum _{l=0}^{k-1}{binom {2k-1}{2l}}{frac {(-1)^{l}A_{2k-2l-1}}{2l+2m+1}}right)-{frac {(-1)^{k-1}}{2m+2k}}right){frac {A_{2m}}{(2m)!}}{left({frac {pi }{2}}right)}^{2m+2k},}$$
where ${displaystyle A_{k}}$ is the Euler zigzag number.
However, the citation is missing on Wikipedia. Is there an easy way to derive this, or alternatively, a link to a paper with a proof?
reference-request special-functions beta-function
asked Nov 12 '18 at 13:25
KlangenKlangen
1,65711334
2
Contour integral, residue theorem, definition of $A_k$ as the Taylor coefficients of a trigonometric function
– reuns
Nov 14 '18 at 16:12
@reuns To the point as usual. If you could (rigorously) expand that thought, I might even throw in some extra points.
– Klangen
Nov 14 '18 at 16:16
2
One source online is vixra.org/pdf/1608.0429v1.pdf, with the desired result following from substituting equation (8) into equation (22). (The similarity of the notation to the Wikipedia page makes me wonder if this is indeed where it was obtained from.)
– Semiclassical
Nov 14 '18 at 16:50
@Semiclassical Thank you, that's interesting. I'm usually very skeptical of vixra papers, has this been published somewhere else?
– Klangen
Nov 15 '18 at 10:01
1
That's my reaction to vixra as well, but I haven't yet been able to find any other source. That being the case, It may be worth posting the relevant derivation from that source (as a CW answer, perhaps) so that other users can directly assess its validity.
– Semiclassical
Nov 15 '18 at 18:56
add a comment |
According to Wikipedia, we have:
$${displaystyle beta (2k)={frac {1}{2(2k-1)!}}sum _{m=0}^{infty }left(left(sum _{l=0}^{k-1}{binom {2k-1}{2l}}{frac {(-1)^{l}A_{2k-2l-1}}{2l+2m+1}}right)-{frac {(-1)^{k-1}}{2m+2k}}right){frac {A_{2m}}{(2m)!}}{left({frac {pi }{2}}right)}^{2m+2k},}$$
where ${displaystyle A_{k}}$ is the Euler zigzag number.
However, the citation is missing on Wikipedia. Is there an easy way to derive this, or alternatively, a link to a paper with a proof?
reference-request special-functions beta-function
asked Nov 12 '18 at 13:25
KlangenKlangen
1,65711334
According to Wikipedia, we have:
$${displaystyle beta (2k)={frac {1}{2(2k-1)!}}sum _{m=0}^{infty }left(left(sum _{l=0}^{k-1}{binom {2k-1}{2l}}{frac {(-1)^{l}A_{2k-2l-1}}{2l+2m+1}}right)-{frac {(-1)^{k-1}}{2m+2k}}right){frac {A_{2m}}{(2m)!}}{left({frac {pi }{2}}right)}^{2m+2k},}$$
where ${displaystyle A_{k}}$ is the Euler zigzag number.
However, the citation is missing on Wikipedia. Is there an easy way to derive this, or alternatively, a link to a paper with a proof?
reference-request special-functions beta-function
reference-request special-functions beta-function
asked Nov 12 '18 at 13:25
KlangenKlangen
1,65711334
asked Nov 12 '18 at 13:25
KlangenKlangen
1,65711334
edited Dec 13 '18 at 10:10
Klangen
asked Nov 12 '18 at 13:25
KlangenKlangen
1,65711334
asked Nov 12 '18 at 13:25
KlangenKlangen
1,65711334
asked Nov 12 '18 at 13:25
KlangenKlangen
1,65711334
1,65711334
2
Contour integral, residue theorem, definition of $A_k$ as the Taylor coefficients of a trigonometric function
– reuns
Nov 14 '18 at 16:12
@reuns To the point as usual. If you could (rigorously) expand that thought, I might even throw in some extra points.
– Klangen
Nov 14 '18 at 16:16
2
One source online is vixra.org/pdf/1608.0429v1.pdf, with the desired result following from substituting equation (8) into equation (22). (The similarity of the notation to the Wikipedia page makes me wonder if this is indeed where it was obtained from.)
– Semiclassical
Nov 14 '18 at 16:50
@Semiclassical Thank you, that's interesting. I'm usually very skeptical of vixra papers, has this been published somewhere else?
– Klangen
Nov 15 '18 at 10:01
1
That's my reaction to vixra as well, but I haven't yet been able to find any other source. That being the case, It may be worth posting the relevant derivation from that source (as a CW answer, perhaps) so that other users can directly assess its validity.
– Semiclassical
Nov 15 '18 at 18:56
add a comment |
2
Contour integral, residue theorem, definition of $A_k$ as the Taylor coefficients of a trigonometric function
– reuns
Nov 14 '18 at 16:12
@reuns To the point as usual. If you could (rigorously) expand that thought, I might even throw in some extra points.
– Klangen
Nov 14 '18 at 16:16
2
One source online is vixra.org/pdf/1608.0429v1.pdf, with the desired result following from substituting equation (8) into equation (22). (The similarity of the notation to the Wikipedia page makes me wonder if this is indeed where it was obtained from.)
– Semiclassical
Nov 14 '18 at 16:50
@Semiclassical Thank you, that's interesting. I'm usually very skeptical of vixra papers, has this been published somewhere else?
– Klangen
Nov 15 '18 at 10:01
1
That's my reaction to vixra as well, but I haven't yet been able to find any other source. That being the case, It may be worth posting the relevant derivation from that source (as a CW answer, perhaps) so that other users can directly assess its validity.
– Semiclassical
Nov 15 '18 at 18:56
2
2
Contour integral, residue theorem, definition of $A_k$ as the Taylor coefficients of a trigonometric function
– reuns
Nov 14 '18 at 16:12
Contour integral, residue theorem, definition of $A_k$ as the Taylor coefficients of a trigonometric function
– reuns
Nov 14 '18 at 16:12
@reuns To the point as usual. If you could (rigorously) expand that thought, I might even throw in some extra points.
– Klangen
Nov 14 '18 at 16:16
@reuns To the point as usual. If you could (rigorously) expand that thought, I might even throw in some extra points.
– Klangen
Nov 14 '18 at 16:16
2
2
One source online is vixra.org/pdf/1608.0429v1.pdf, with the desired result following from substituting equation (8) into equation (22). (The similarity of the notation to the Wikipedia page makes me wonder if this is indeed where it was obtained from.)
– Semiclassical
Nov 14 '18 at 16:50
One source online is vixra.org/pdf/1608.0429v1.pdf, with the desired result following from substituting equation (8) into equation (22). (The similarity of the notation to the Wikipedia page makes me wonder if this is indeed where it was obtained from.)
– Semiclassical
Nov 14 '18 at 16:50
@Semiclassical Thank you, that's interesting. I'm usually very skeptical of vixra papers, has this been published somewhere else?
– Klangen
Nov 15 '18 at 10:01
@Semiclassical Thank you, that's interesting. I'm usually very skeptical of vixra papers, has this been published somewhere else?
– Klangen
Nov 15 '18 at 10:01
1
1
That's my reaction to vixra as well, but I haven't yet been able to find any other source. That being the case, It may be worth posting the relevant derivation from that source (as a CW answer, perhaps) so that other users can directly assess its validity.
– Semiclassical
Nov 15 '18 at 18:56
That's my reaction to vixra as well, but I haven't yet been able to find any other source. That being the case, It may be worth posting the relevant derivation from that source (as a CW answer, perhaps) so that other users can directly assess its validity.
– Semiclassical
Nov 15 '18 at 18:56
add a comment |
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2
Contour integral, residue theorem, definition of $A_k$ as the Taylor coefficients of a trigonometric function
– reuns
Nov 14 '18 at 16:12
@reuns To the point as usual. If you could (rigorously) expand that thought, I might even throw in some extra points.
– Klangen
Nov 14 '18 at 16:16
2
One source online is vixra.org/pdf/1608.0429v1.pdf, with the desired result following from substituting equation (8) into equation (22). (The similarity of the notation to the Wikipedia page makes me wonder if this is indeed where it was obtained from.)
– Semiclassical
Nov 14 '18 at 16:50
@Semiclassical Thank you, that's interesting. I'm usually very skeptical of vixra papers, has this been published somewhere else?
– Klangen
Nov 15 '18 at 10:01
1
That's my reaction to vixra as well, but I haven't yet been able to find any other source. That being the case, It may be worth posting the relevant derivation from that source (as a CW answer, perhaps) so that other users can directly assess its validity.
– Semiclassical
Nov 15 '18 at 18:56