Rectangular rotation of spherical harmonic expansion of arbitrary high degree and order

In order to move the polar singularity of arbitrary spherical harmonic expansion to a point on the equator, we rotate the expansion around the y-axis by $$90^{circ }$$ such that the x-axis becomes a new pole. the expansion coefficients are transformed by multiplying a special value of wigner d-matrix and a normalization factor. the transformation matrix is unchanged whether the coefficients are $$4 pi $$ fully normalized or schmidt quasi-normalized. the matrix is recursively computed by the so-called x-number formulation (fukushima in j geodesy 86: 271–285, 2012a). as an example, we obtained $$2190times 2190$$ coefficients of the rectangular rotated spherical harmonic expansion of egm2008. a proper combination of the original and the rotated expansions will be useful in (i) integrating the polar orbits of artificial satellites precisely and (ii) synthesizing/analyzing the gravitational/geomagnetic potentials and their derivatives accurately in the high latitude regions including the arctic and antarctic area.