  The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC)
stability polynomials of arbitrary order $N$ is presented. Roots of FRKC
stability polynomials of degree $L=MN$ are used to construct explicit schemes
comprising $L$ forward Euler stages with internal stability ensured through a
sequencing algorithm which limits the internal amplification factors to $\sim
L^2$. The associated stability domain scales as $M^2$ along the real axis.
Marginally stable real-valued points on the interior of the stability domain
are removed via a prescribed damping procedure.
  By construction, FRKC schemes meet all linear order conditions; for nonlinear
problems at orders above 2, complex splitting or Butcher series composition
methods are required. Linear order conditions of the FRKC stability polynomials
are verified at orders 2, 4, and 6 in numerical experiments. Comparative
studies with existing methods show the second-order unsplit FRKC2 scheme and
higher order (4 and 6) split FRKCs schemes are efficient for large moderately
stiff problems.
