Hypergeometric-type sequences

We introduce hypergeometric-type sequences. They are linear combinations of interlaced hypergeometric sequences (of arbitrary interlacements). We prove that they form a subring of the ring of holonomic sequences. An interesting family of sequences in this class are those defined by trigonometric functions with linear arguments in the index and π, such as Chebyshev polynomials, ${({\sin }^2(n\pi /4)\cdot \cos (n\pi /6))}_n$, and compositions like ${(\sin (\cos (n\pi /3\left)\pi \right))}_n$.

We describe an algorithm that computes a hypergeometric-type normal form of a given holonomic nth term whenever it exists. Our implementation enables us to generate several identities for terms defined via trigonometric functions.