Gosper's algorithm and Bell numbers

TL;DR

This paper explores how multiplying certain hypergeometric summands by specific polynomials, described via Bell numbers, can enable closed-form evaluation of finite sums that are otherwise intractable.

Contribution

It provides an explicit description of polynomials, related to Bell numbers, that fix non-closed-form hypergeometric sums, extending Gosper's algorithm.

Findings

Identifies a class of polynomials that transform sums into closed forms

Connects Bell numbers with hypergeometric summation techniques

Offers explicit formulas for polynomial fixing factors

Abstract

Computers are good at evaluating finite sums in closed form, but there are finite sums which do not have closed forms. Summands which do not produce a closed form can often be ``fixed'' by multiplying them by a suitable polynomial. We provide an explicit description of a class of such polynomials for simple hypergeometric summands in terms of the Bell numbers.