Ramanujan’s Master theorem states that (under certain conditions) if a function $f$ on $\mathbb R$ can be expanded around zero in a power series of the form \begin{equation} f(x)=\sum_{k=0}^\infty (-1)^k a(k) x^k, \end{equation} then \begin{equation} \int_0^\infty f(x) x^{-\lambda-1}\,dx=-\frac{\pi}{\sin\pi\lambda} a(\lambda), \, \text{ for }\lambda\in\mathbb C. \end{equation} This theorem can be thought of as an interpolation theorem, which reconstructs the values of $a(\lambda)$ from its given values at $a(k), k\in \mathbb N\cup {0}$. In particular if $a(k)=0$ for all $k\in \mathbb N\cup {0}$, then $a$ is identically $0$. By selecting particular values for the function $a$, Ramanujan applied this theorem to compute several definite integrals and power series. This explains why it is referred to as the “Master Theorem”.

Based on the duality of Riemannian symmetric spaces of compact and noncompact type inside a common complexification, Bertram, Olafsson-Pasquale proved an analogue of this theorem on Riemannian symmetric spaces of noncompact type.

In the first part of the talk, we shall discuss an analogue of this theorem for radial sections of line bundles over Poincare upper half plane. This is a joint work with Swagato K Ray.

In the second half, we shall discuss an analogue of this theorem for Sturm-Liouville operators. This is a joint work with Jotsaroop Kaur.

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Last updated: 04 Dec 2020