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Ramanujan's Interpolation Formula

Ramanujan's interpolation formula relates integrals of power series to values of their coefficient functions. Let phi(n) be any function, say analytic or integrable. Then

int_0^inftyx^(s-1)sum_(k=0)^infty(-1)^kx^kphi(k)dx=(piphi(-s))/(sin(spi))
(1)

and

int_0^inftyx^(s-1)sum_(k=0)^infty(-1)^k(x^k)/(k!)lambda(k)dx=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

phi(u)=(lambda(u))/(Gamma(1+u)).
(3)

These formulas give valid results only for certain classes of functions, and are connected with Mellin transforms (Hardy 1999, p. 15).

See also

Ramanujan's Master Theorem

Portions of this entry contributed by Jonathan Sondow (author's link)

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References

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 15 and 186-195, 1999.

Referenced on Wolfram|Alpha

Ramanujan's Interpolation Formula

Cite this as:

Weisstein, Eric W., with contributions by Jonathan Sondow. "Ramanujan's Interpolation Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RamanujansInterpolationFormula.html

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