Polynomial reduction for $q$-holonomic sequences

Abstract

This paper provides a (Laurent) polynomial reduction to $q$-holonomic sequences $F_k(q)$. We first characterize Laurent polynomials $\tilde{p}(x)$ such that the product $\tilde{p}(q^k)F_k(q)$ is summable. Then the reduction framework is given to decompose any given Laurent polynomial into a summable part and a remainder with lower degree. Finally, we introduce a power-partible reduction for $q$-holonomic sequences of which the recurrence relation satisfies a certain symmetry condition. The advantage is that it can not only simultaneously eliminate the highest-degree and lowest-degree terms of a Laurent polynomial satisfying a symmetry condition, but also guarantee the symmetry of the remainder. As applications, we apply the reduction to $q$-central-Delannoy numbers to derive new $q$-identities and $q$-congruences.