This site is supported by donations to The OEIS Foundation.
Dirichlet beta function
From OeisWiki
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
where
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
Contents
Formulae
where are the Euler numbers.
where is the polygamma function[2], are the Bernoulli numbers and the Riemann zeta function for even integers is given by
See also
Notes
- ↑ 1.0 1.1 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.
- ↑ The polygamma function is the -th derivative of the digamma function (which is the logarithmic derivative of the Gamma function ).
External links
- Weisstein, Eric W., Dirichlet Beta Function, from MathWorld—A Wolfram Web Resource.