  A new generalization of curvature modified plasma dispersion functions is
introduced in order to express Dupree renormalized dispersion relations used in
quasi-linear theory. For instance the Dupree renormalized dispersion relation
for gyrokinetic, toroidal ion temperature gradient driven (ITG) modes, where
the Dupree's diffusion coefficient is assumed to be a low order polynomial of
the velocity, can be written entirely using generalized curvature modified
plasma dispersion functions: Knm's. Using those, Dupree's formulation of
renormalized quasi-linear theory is revisited for the toroidal ITG mode. The
Dupree diffusion coefficient has been obtained as a function of velocity using
an iteration scheme, first by assuming that the diffusion coefficient is
constant at each v (i.e. applicable for slow dependence), and then substituting
the resulting v dependence in the form of complex polynomial coefficients into
the Knm's for verification. The algorithm generally converges rapidly after
only a few iterations. Since the quasi-linear calculation relies on an assumed
form for the wave-number spectrum, especially around its peak, practical
usefullness of the method is to be determined in actual applications.
