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We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups  USp (2 N ),  SO (2 N ) and  O  − (2 N ). These averages are used to predict the asymptotic formulae for moments of derivatives of  L -functions which arise in number theory. Each formula gives the leading constant of the asymptotic in terms of determinants of hypergeometric functions. We find a differential recurrence relation between these determinants that allows the rapid computation of the ( k +1)st constant in terms of the  k th and ( k −1)st. This recurrence is reminiscent of a Toda lattice equation arising in the theory of τ-functions associated with Painlevé differential equations.

A RECURSION FORMULA FOR MOMENTS OF DERIVATIVES OF RANDOM MATRIX POLYNOMIALS