Ramanujan's Interpolation Formula

$\displaystyle \int_0^\infty x^{s-1} \sum_{k=0}^\infty (-1)^kx^k\phi(k) \,dx$	$\textstyle =$	$\displaystyle {\pi\phi(-s)\over\sin(s\pi)}$	(1)
$\displaystyle \int_0^\infty x^{s-1} \sum_{k=0}^\infty (-1)^k {x^k\over k!}\lambda(k)\,dx$	$\textstyle =$	$\displaystyle \Gamma(s)\lambda(-s),$	(2)

where $\lambda(z)$ is the Dirichlet Lambda Function and $\Gamma(z)$ is the Gamma Function. Equation (2) is obtained from (1) by defining

$\begin{displaymath} \phi(u)={\lambda(u)\over\Gamma(1+u)}. \end{displaymath}$

(3)

These formulas give valid results only for certain classes of functions.

References