A Computer Algebra Package for Polynomial Sequence Recognition

TL;DR

This paper introduces a specialized computer algebra package that uses a hybrid, factorization-based approach to automatically recognize and generate formulas for polynomial sequences, including those involving special sequences like Stirling numbers.

Contribution

It presents a novel, factorization-based method for polynomial sequence recognition that integrates user guidance with existing computational tools, enabling recognition of complex sequences not handled by other software.

Findings

Successfully recognizes polynomial sequences involving Stirling numbers

Combines user input with existing functions for sequence recognition

Demonstrates capabilities through concrete working examples

Abstract

The software package developed in the MS thesis research implements functions for the intelligent guessing of polynomial sequence formulas based on user-defined expected sequence factors of the input coefficients. We present a specialized hybrid approach to finding exact representations for polynomial sequences that is motivated by the need for an automated procedures to discover the precise forms of these sums based on user guidance, or intuition, as to special sequence factors present in the formulas. In particular, the package combines the user input on the expected special sequence factors in the polynomial coefficient formulas with calls to the existing functions as subroutines that then process formulas for the remaining sequence terms already recognized by these packages. The factorization-based approach to polynomial sequence recognition is unique to this package and allows the search functions to find expressions for polynomial sums involving Stirling numbers and other special triangular sequences that are not readily handled by other software packages. In contrast to many other sequence recognition and summation software, the package not provide an explicit proof, or certificate, for the correctness of these sequence formulas -- only computationally guided educated guesses at a complete identity generating the sequence over all $n$. The thesis contains a number of concrete, working examples of the package that are intended to both demonstrate its usage and to document its current sequence recognition capabilities.