Ramanujanʹs Master Theorem and Duality of Symmetric Spaces

We interpret Ramanujanʹs Master Theorem (B. Berndt, “Ramanujanʹs Notebooks, Part I,” Springer-Verlag, New York, 1985),∫∞0 x−s−1 ∑k=0∞ ((−1)k a(k) xk) dx=−πsin(πs) a(s) (R)as a relation between the Fourier transforms of an analytic functionfwith respect to the real formsU(1) (compact) and R+(non-compact) of the multiplicative group of non-zero complex numbers, and we ask for a similar relation between the spherical Fourier transforms of an analytic function with respect to a compact real form and the non-compact dual real form of a complex symmetric space. We obtained results in the case of symmetric cones and in the rank-one case. Here we present the latter case in detail, describing features which will be also important for the general rank case