  In this paper we extend the polynomial time integration framework to include
exponential integration for both partitioned and unpartitioned initial value
problems. We then demonstrate the utility of the exponential polynomial
framework by constructing a new class of parallel exponential polynomial block
methods (EPBMs) based on the Legendre points. These new integrators can be
constructed at arbitrary orders of accuracy and have improved stability
compared to existing exponential linear multistep methods. Moreover, if the ODE
right-hand side evaluations can be parallelized efficiently, then high-order
EPBMs are significantly more efficient at obtaining highly accurate solutions
than exponential linear multistep methods and exponential spectral deferred
correction methods of equivalent order.
