How is this formula for the Dirichlet $beta$-function derived?
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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 beta-function
asked 19 hours ago
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up vote
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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 beta-function
asked 19 hours ago
Flermat
1,20911129
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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 beta-function
asked 19 hours ago
Flermat
1,20911129
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 beta-function
reference-request beta-function
asked 19 hours ago
Flermat
1,20911129
asked 19 hours ago
Flermat
1,20911129
edited 18 hours ago
asked 19 hours ago
Flermat
1,20911129
asked 19 hours ago
Flermat
1,20911129
asked 19 hours ago
Flermat
1,20911129
1,20911129
add a comment |
add a comment |
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