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A rigorous mathematical analysis is given of spherical stacked-patch arrays with emphasis on the physical interpretation of mutual coupling mechanisms present in doubly- curved convex structures. The analysis method is based on electromagnetic field representation in terms of spherical harmonics where each harmonic has the same angular variation as the spectral source component. To obtain the spectral representation the vector-Legendre transformation is applied to currents and fields. A novel approach to the mutual coupling calculation within the method of moments analysis of spherical arrays is applied. By expressing the patch current in terms of two suitable potential-like auxiliary functions, it is possible to avoid the use of Euler's formulas for coordinate system rotation and the related lengthy integrations. Instead, the rotation of antenna elements and corresponding current distributions can be done in closed form with the help of Vilenkin's addition theorem for associated Legendre functions. It is shown that the new approach results in significant acceleration and improved accuracy of the analysis of spherical patch antenna arrays. The algorithm is successfully tested against a commercially available electromagnetic software and measurements performed on the developed laboratory model, confirming its accuracy for both input impedance and mutual coupling calculation and with only a small difference between the predicted and measured resonant frequencies, due to limitations in the experimental model. The influence of the structure parameters on mutual coupling level is extensively investigated, including all coupling mechanisms and leakage of energy due to curvature of the structure. It is shown that stacked-patch antennas can have reduced coupling level comparing to single patch antennas with possible deep nulls above the antenna resonant frequency.