Summary: In rapidly rotating spheres, the whole fluid column, extending from the southern to northern spherical boundary along the rotation axis, moves like a single fluid element, which is usually referred to as geostrophic flow. A new Legendre-type polynomial is discovered in undertaking the asymptotic analysis of geostrophic flow in spherical geometry. Three essential properties characterize the new polynomial: (i) it is a function of \(r\) and \(\theta \) but takes a single argument \((r\sin \theta)\), which is restricted by \(0\leq r\leq 1\) and \(0\leq \theta\leq \pi \), where \((r,\theta ,\phi )\) denote spherical polar coordinates with \(\theta =0\) at the rotation axis; (ii) it is odd and vanishes at the axis of rotation \(\theta =0\), and (iii) it is defined within-and orthogonal over-the full sphere. As an example of its application, we employ the new polynomial in the asymptotic analysis of forced geostrophic flows in rotating fluid spheres for small Ekman and Rossby numbers. Fully numerical analysis of the same problem is also carried out, showing satisfactory agreement between the asymptotic solution and the numerical solution.