  In this paper, we present a new class of operators, which we name to be
$n$-Ritt operators. This produces a discrete analogue of $n$-sectorial
operators and generalizes the notion of Ritt operators. We develop a
$H^\infty$-functional calculus for $n$-Ritt operators and prove an useful
transference result. We also generalize the notions of quadratic functional
calculus and $n$-$R$-Ritt operator and discuss some examples.
