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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
\end{displaymath} (3)

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


Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 15 and 186-195, 1959.

© 1996-9 Eric W. Weisstein