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Using results on multiply monotone functions, we establish the positivity of integrals of Bessel functions of the form \[ \int_0^x {\left( {x^\mu - t^\mu } \right)^\lambda } t^\alpha J_\beta (t)dt,\quad x > 0,\] where $0 < \mu \leq 1 \leq \lambda $ and $\alpha $, $\beta $ satisfy various conditions. In particular, the result holds if $ - \frac{1}{2} \leq \alpha = \beta \leq \frac{3}{2}$ or if $\frac{3}{2} = \alpha \leq \beta $.

Results are obtained for the problems of constructing and characterizing scalar-valued polynomial statistics having constant regression on the mean of a random sample of Wishart matrices. The construction procedure introduced by Heller [J. Multivariate Anal., 14 (1984), pp. 101–104] is generalized to show that certain polynomials in the principal minors of the sample matrices have zero regression on the mean. The zero-regression polynomials are characterized through expectations involving certain matrix-valued Bessel functions of Gross and Kunze [J. Funct. Anal., 22 (1976), pp. 73–105]. It is shown that the zero-regression property characterizes Wishart distributions within a wide family of mixtures of Wishart distributions.

It is shown that the hypergeometric function of $m \times m$ scalar matrix argument (cf. Herz, Annals of Math., 61 (1955), pp. 474–523) may be expressed as the Pfaffian of a matrix whose entries are evaluated in terms of classical hypergeometric functions. Applications, in multivariate statistical theory, are made to the distributions of eigenvalues of various random matrices.