Hypergeometric-Type Sequences

Abstract

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 $\pi$, such as Chebyshev polynomials, ${\left({\sin }^2\left(n\pi /4\right)\cdot \cos \left(n\pi /6\right)\right)}_n$, and compositions like ${\left(\sin \left(\cos (n\pi /3)\pi \right)\right)}_n$. We describe an algorithm that computes a hypergeometric-type normal form of a given holonomic $n\text{th}$ term whenever it exists. Our implementation enables us to generate several identities for terms defined via trigonometric functions.

23 pages. To appear in the Journal of Symbolic Computation