  We consider the problem of calculating distance correlation coefficients
between random vectors whose joint distributions belong to the class of
Lancaster distributions. We derive under mild convergence conditions a general
series representation for the distance covariance for these distributions. To
illustrate the general theory, we apply the series representation to derive
explicit expressions for the distance covariance and distance correlation
coefficients for the bivariate normal distribution and its generalizations of
Lancaster type, the multivariate normal distributions, and the bivariate gamma,
Poisson, and negative binomial distributions which are of Lancaster type.
