  The circular and Jacobi ensembles of random matrices have their eigenvalue
support on the unit circle of the complex plane and the interval $(0,1)$ of the
real line respectively. The averaged value of the modulus of the corresponding
characteristic polynomial raised to the power $2 \mu$ diverges, for $2\mu \le
-1$, at points approaching the eigenvalue support. Using the theory of
generalized hypergeometric functions based on Jack polynomials, the functional
form of the leading asymptotic behaviour is established rigorously. In the
circular ensemble case this confirms a conjecture of Berry and Keating.
