\[f(t)=\frac{1}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}%
\widetilde{f}(\mathrm{i}\lambda)\Phi^{(\alpha,\beta)}_{\mathrm{i}\lambda}(t)%
\frac{\Gamma\left(\tfrac{1}{2}(\alpha+\beta+1+\lambda)\right)\Gamma\left(%
\tfrac{1}{2}(\alpha-\beta+1+\lambda)\right)}{\Gamma\left(\alpha+1\right)\Gamma%
\left(\lambda\right)2^{\alpha+\beta+1-\lambda}}\,\mathrm{d}\lambda,\]