Control of cancellations that restrain the growth of a binomial recursion

TL;DR

This paper investigates a parameter-dependent recursion with cancellations, revealing factorial growth for most parameters and exponential growth at a specific point, using integral operators and combinatorial analysis.

Contribution

It introduces a novel approach linking the recursion to integral operators and characterizes growth behavior across parameter values.

Findings

Sequence growth is factorial for all parameters except -1.

At x = -1, the sequence corresponds to alternating Catalan numbers with exponential growth.

The methods may apply to similar recursive sequences with cancellations.

Abstract

We study a recursion that generates real sequences depending on a parameter $x$. Given a negative $x$ the growth of the sequence is very difficult to estimate due to canceling terms. We reduce the study of the recursion to a problem about a family of integral operators, and prove that for every parameter value except -1, the growth of the sequence is factorial. In the combinatorial part of the proof we show that when $x=-1$ the resulting recurrence yields the sequence of alternating Catalan numbers, and thus has exponential growth. We expect our methods to be useful in a variety of similar situations.