An extension of holonomic sequences: C^2-finite sequences

Holonomic sequences are widely studied as many objects interesting to mathematicians and computer scientists are in this class. In the univariate case, these are the sequences satisfying linear recurrences with polynomial coefficients and also referred to as $D$-finite sequences. A subclass are $C$-finite sequences satisfying a linear recurrence with constant coefficients. We investigate the set of sequences which satisfy linear recurrence equations with coefficients that are $C$-finite sequences. These sequences are a natural generalization of holonomic sequences. In this paper, we show that $C^2$-finite sequences form a difference ring and provide methods to compute in this ring. Furthermore, we provide an analogous construction for $D^2$-finite sequences, i.e., sequences satisfying a linear recurrence with holonomic coefficients. We show that these constructions can be iterated and obtain an increasing chain of difference rings.