Constructing minimal telescopers for rational functions in three discrete variables

We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed by a known existence criterion of telescopers. Our approach has the important feature that it avoids the potentially costly computation of certificates. Computational experiments are also provided so as to illustrate the efficiency of our approach

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, sin2 (n π/4) · cos (n π/6))n , and compositions like (sin (cos(nπ/3)π))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 function