FPS In Action: An Easy Way To Find Explicit Formulas For Interlaced Hypergeometric Sequences

TL;DR

This paper demonstrates how recent updates to the Formal Power Series algorithm can efficiently find explicit, simple formulas for interlaced hypergeometric sequences by computing their generating functions, aiding sequence analysis.

Contribution

The paper introduces an improved FPS algorithm implementation that effectively derives explicit formulas for P-recursive sequences, enhancing sequence analysis tools.

Findings

Successfully finds explicit formulas for complex sequences

Implements in Maxima and Maple for practical use

Enhances understanding of hypergeometric sequences

Abstract

Linear recurrence equations with constant coefficients define the power series coefficients of rational functions. However, one usually prefers to have an explicit formula for the sequence of coefficients, provided that such a formula is "simple" enough. Simplicity is related to the compactness of the formula due to the presence of algebraic numbers: "the smaller, the simpler". This poster showcases the capacity of recent updates on the Formal Power Series (FPS) algorithm, implemented in Maxima and Maple (convert/FormalPowerSeries), to find simple formulas for sequences like those from https://oeis.org/A307717, https://oeis.org/A226782, or https://oeis.org/A226784 by computing power series representations of their correctly guessed generating functions. We designed the algorithm for the more general context of univariate $P$-recursive sequences. Our implementations are available at http://www.mathematik.uni-kassel.de/~bteguia/FPS_webpage/FPS.htm