Hypergeometric-Type Sequences
Bertrand Teguia Tabuguia
TL;DR
The paper formalizes hypergeometric-type sequences as finite linear combinations of interlaced $m$-fold hypergeometric terms, situating them inside the holonomic ($P$-recursive) framework. It establishes that these sequences form a ring and are generated by proper hypergeometric-type power series, with a tight link between discrete hypergeometric-type terms and generating functions. An algorithmic pipeline is developed to convert a given holonomic term into a hypergeometric-type normal form: derive a holonomic recurrence, extract a basis of $m$-fold hypergeometric solutions, and solve a linear system against initial values to obtain the normal form; this is implemented in Maple as the HyperTypeSeq package (HolonomicRE, REtoHTS, HTS). The work enables automated discovery of closed forms and identities for trig- and elliptic-function-based sequences, and points to future generalizations to Puiseux series and broader connections to C-finite sequences and holonomic theory.
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 $π$, such as Chebyshev polynomials, $\left(\sin^2\left(n\,π/4\right)\cdot\cos\left(n\,π/6\right)\right)_n$, and compositions like $\left(\sin\left(\cos(nπ/3)π\right)\right)_n$. 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.