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We give asymptotic formulae for random matrix averages of derivatives ofcharacteristic polynomials over the groups USp(2N), SO(2N) and O^-(2N). Theseaverages are used to predict the asymptotic formulae for moments of derivativesof L-functions which arise in number theory. Each formula gives the leadingconstant of the asymptotic in terms of determinants of hypergeometricfunctions. We find a differential recurrence relation between thesedeterminants which allows the rapid computation of the (k+1)-st constant interms of the k-th and (k-1)-st. This recurrence is reminiscent of a Todalattice equation arising in the theory of \tau-functions associated withPainlev\'e differential equations.

A Recursion Formula for Moments of Derivatives of Random Matrix Polynomials