Using renormalization techniques on the Englert linked-cluster expansion we have derived high-temperature series for the spin-spin correlation function of the three-dimensional, plane-rotator, and spin-infinity XY models with nearest-neighbor interactions. Analysis of our zero-field series, which are two terms longer than earlier series for these models, gives results in basic agreement with earlier work. Our finite-field series, to eighth order in the interaction and second order in the field, allow a determination of the gap index Δ; we assert 2Δ=3.33±0.02. The indices are compared with scaling predictions and experimental results. Accessible critical amplitudes are determined. These are used to test predictions of two-scale-factor universality; good agreement is observed.