Computing with D-Algebraic Sequences

A sequence is difference algebraic (or D-algebraic) if finitely many shifts of its general term satisfy a polynomial relationship; that is, they are the coordinates of a generic point on an affine hypersurface. The corresponding equations are called algebraic difference equations (ADE). We show that subsequences of D-algebraic sequences, indexed by arithmetic progressions, satisfy ADEs of the same orders as the original sequences. Additionally, we provide algorithms for operations with D-algebraic sequences and discuss the difference-algebraic nature of holonomic and $C^2$-finite sequences.