Ramanujan’s master theorem

Ramanujan's passport photo from 1913

A few weeks ago I wrote about the Mellin transform. Mitchell Wheat left comment saying the transform seems reminiscent of Ramanujan’s master theorem, which motivated this post.

Suppose you have a function f that is nice enough to have a power series.

$f(z) = \sum_{k=0}^\infty a_k z^k$

Now focus on the coefficients a_k as a function of k. We’ll introduce a function φ that yields the coefficients, with a twist.

$f(z) = \sum_{k=0}^\infty \frac{\varphi(k)}{k!} (-z)^k$

and so φ(k) = (−1)^k k! a_k. Another way to look at it is that f is the exponential generating function of (−1)^k φ(k).

Then Ramanujan’s master theorem gives a formula for the Mellin transform of f:

$\int_0^\infty z^{s-1} f(z) \, dz = \Gamma(s) \varphi(-s)$

This equation was the basis of many of Ramanujan’s theorems.

Master theorem of algorithm analysis
Ramanujan’s approximation for the perimeter of an ellipse
Transforms and convolutions

Ramanujan’s master theorem

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