Polynomial reduction for holonomic sequences and applications in $\pi$-series and congruences

TL;DR

This paper extends polynomial reduction techniques from hypergeometric terms to holonomic sequences, enabling automated proof and generation of multi-summation identities, including new $$-series involving special number sequences.

Contribution

It introduces a novel polynomial reduction method for holonomic sequences, facilitating automated identity proofs and generating new $$-series with applications to special number sequences.

Findings

New algorithms for proving multi-summation identities

Discovery of novel $$-series involving Domb and Franel numbers

Extension of polynomial reduction to holonomic sequences

Abstract

Polynomial reduction, designed first for hypergeometric terms, can be used to automatically prove and generate new hypergeometric identities from old ones. In this paper, we extend the reduction method to holonomic sequences. As applications, we describe an algorithmic way to prove and generate new multi-summation identities. Especially we present new families of $\pi$-series involving Domb numbers and Franel numbers.