A class of alternatives is here presented to Fisher's (1925) expansion of Student's $t$ density function. These expansions involve Appell's polynomials; and hence, recurrence schemes are available for the coefficients. Complete integrals of products of $t$ densities are of interest as Behrens-Fisher densities (viewed as Bayesian posterior distributions: Jeffreys, 1940; Patil, 1964) as moments of Bayesian posterior distributions (Anscombe, 1963; Tiao and Zellner, 1964). A symptotic expansion of complete integrals, obtained by term-by-term integration of these expansions, are favorably compared with those obtained from Fisher's expansion. Although expansions of complete integrals of products of multivariate $t$ densities can be developed from these expansions by the methods of Tiao and Zellner, the resulting coefficients are practically as complicated as the Tiao and Zellner coefficients; methods will be published soon (Dickey, 1965) for reducing the dimensionality of such integrals for quadrature. The paper concludes with a numerical study of the integral expansions.