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Dirichlet beta function

The Dirichlet beta function (also known as the Catalan beta function) is a special function closely related to the Riemann zeta function. It is defined as

${\displaystyle \beta (s):=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{s}}}=\sum _{n=1}^{\infty }{\frac {{\chi _{\beta }}(n)}{n^{s}}},\quad s>0,\,}$

where

${\displaystyle {\chi _{\beta }}(n)=\cos \,{\frac {(n-1)\pi }{2}},\quad n\geq 1,\,}$

is the alternating character (of period four) of the above Dirichlet L-function, giving the sequence (A056594)

{1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...}

with generating function

${\displaystyle G_{\{{\chi _{\scriptscriptstyle \beta }}(n)\}}(x)={\frac {x}{1+x^{2}}},\quad n\geq 1.\,}$

Formulae

${\displaystyle \beta (2n+1)=(-1)^{n}\,{\frac {E_{2n}}{2\,(2n)!}}\left({\frac {\pi }{2}}\right)^{2n+1},\quad n\geq 0,\,}$ [1]

where ${\displaystyle \scriptstyle E_{n}\,}$ are the Euler numbers.

${\displaystyle \beta (2n)={\frac {\psi ^{(2n-1)}({\frac {1}{4}})}{2\,(2n-1)!\,4^{2n-1}}}-{\frac {(2^{2n-1}-{\frac {1}{2}})\,|B_{2n}|\,\pi ^{2n}}{(2n)!}}={\frac {\psi ^{(2n-1)}({\frac {1}{4}})}{2\,(2n-1)!\,4^{2n-1}}}-(1-2^{-2n})\,\zeta (2n),\quad n\geq 1,\,}$ [1]

where ${\displaystyle \scriptstyle \psi ^{(n)}(x)\,}$ is the polygamma function[2], ${\displaystyle \scriptstyle B_{n}\,}$ are the Bernoulli numbers and the Riemann zeta function for even integers is given by

${\displaystyle \zeta (2n)={\frac {2^{2n-1}\,|B_{2n}|\,\pi ^{2n}}{(2n)!}},\quad n\geq 1.\,}$

• Catalan's constant (${\displaystyle \scriptstyle \beta (2)\,}$)
1. F. M. S. Lima, An Euler-type formula for $β(2n)$ and closed-form expressions for a class of zeta series, arXiv:0910.5004, 2009, 2011.
2. The polygamma function ${\displaystyle \scriptstyle \psi ^{(n)}(x)\,}$ is the ${\displaystyle \scriptstyle n\,}$-th derivative of the digamma function ${\displaystyle \scriptstyle \psi (x)\,}$ (which is the logarithmic derivative of the Gamma function ${\displaystyle \scriptstyle \Gamma (x)\,}$).