Some identities for spheroidal harmonics

The spheroidal harmonics expressions $$left[ {P_{2k}^{2s} left( {ixi } right)P_{2k - 2r}^{2s} left( eta right) - P_{2k - 2r}^{2s} left( {ixi } right)P_{2k}^{2s} left( eta right)} right]e^{i2stheta } $$ and $$left[ {eta ^2 P_{2k}^{2s} left( {ixi } right)P_{2k - 2r}^{2s} left( eta right) + xi ^2 P_{2k - 2r}^{2s} left( {ixi } right)P_{2k}^{2s} left( eta right)} right]e^{i2stheta } $$ , have ξ²+η² as a factor. A method is presented for obtaining for these two expressions the coefficient of ξ²+η² in the form of a linear combination of terms of the formP _2m ^2s (iξ)P _2n ^2s (η)e ^i2sθ. Explicit formulae are exhibited for the casesr=1, 2, 3 and any positive or zero integersk ands. Such identities are useful in gravitational potential theory for ellipsoidal distributions when matching Legendre function expansions are employed.