Ramanujan’s Master Theorem says that for functions \(a\) in a suitable Hardy space of the closed right half-plane in \(\mathbb R^2\), the following statement holds: \[ \int_0^\infty x^{-s-1} \left(\sum_{k=0}^\infty (-1)^k a(k) x^k \right) \ dx= -\frac{\pi}{\sin(\pi s)} a(s), \quad s \in \mathbb C. \] In the article [J. Funct. Anal. 148, No. 1, 117–151 (1997; Zbl 0880.43012)], W. Bertram puts this Master Theorem of Ramanujan in a new perspective as a statement connecting the spherical Fourier transform on a compact symmetric space and its non-compact dual symmetric space for suitable functions. More precisely, a compact symmetric space \(X_U=U/K\) and its non-compact dual \(X_G=G/K\) can both be realized as real forms of their common complexification \(X_{\mathbb C} =U_{\mathbb C}/K_{\mathbb C}.\) If \(f\) is an analytic \(K_{\mathbb C}\)-invariant function on \(X_{\mathbb C}\), and has “nice” properties, then, as the spherical Fourier transform is invertible, \(f\) can be determined by its non-compact spherical Fourier transform \(\tilde{f}\) (defined for the restriction of \(f\) to \(X_G\)) as well as by its compact spherical Fourier transform \(\hat{f}\) (defined for the restriction of \(f\) to \(X_U\)), which in turn means that these transforms determine each other. In the above cited article, W. Bertram made explicit this relation proving analogues (two versions) of Ramanujan’s Master Theorem for Riemannian symmetric spaces of rank one and many other interesting results on the way.
If we let \(X_U=U(1), X_G={\mathbb R}^+\) then both can be realized as the real form of their complexification \(X_{\mathbb C}={\mathbb C}^{\ast}\). Further \(x^s\), \(s \in \mathbb C\), are spherical functions for the multiplicative group \(\mathbb R^+\) and \(x^m\), \(m \in \mathbb Z\), are spherical functions of \(U(1).\) Therefore a reformulation of the above stated Ramanujan’s Master Theorem in Bertram’s words would say that for \(a\) as before, if we let \(f=\sum_{k=0}^\infty (-1)^k a(k) x^k,\) then \(\tilde{f}(s)=-\frac{\pi}{\sin(\pi s)}a(s)\) and \(\hat{f}(m)=(-1)^m a_m.\)
Later on, G. Ólafsson and A. Pasquale extended this result on Riemannian symmetric spaces of arbitrary rank [J. Funct. Anal. 262, No. 11, 4851–4890 (2012; Zbl 1246.43005)] and for the hypergeometric Fourier transform associated with root systems [G. Ólafsson and A. Pasquale, J. Fourier Anal. Appl. 19, No. 6, 1150–1183 (2013; Zbl 1314.43002)].
In the present article, the authors revisit the rank one case for \(U=SU(2)\), \(K=SO(2)\), \(G=SL(2,\mathbb R)\), but for functions on \(G\) which transform according to the same non-trivial \(K\)-type on both the left and the right. More precisely, for a non-zero integer \(n\), the functions which transform according to type \(n\) on both the left and the right on \(G\) under the \(K\)-action are being considered. The authors then prove a version of Ramanujan’s Master Theorem in this setting following the techniques from the spherical case. The results then reflect some natural changes from the spherical case, and thus answer one of the questions posed by Bertram in his above cited article about Ramanujan’s Master Theorem for \(K\)-finite functions in this special case.