Hankel determinants of middle binomial coefficients and conjectures for some polynomial extensions and modifications

TL;DR

This paper investigates the Hankel determinants of middle binomial coefficients, extends them to polynomial forms, and explores their properties, conjectures, and q-analogs, revealing interesting modular patterns.

Contribution

It provides explicit formulas for Hankel determinants of middle binomial coefficients and introduces polynomial extensions with conjectures and q-analogs.

Findings

Explicit formulas for Hankel determinants and generating functions.

Conjectures on polynomial extensions of middle binomial coefficients.

Observation of modular patterns in modified coefficients.

Abstract

The middle binomial coefficients can be interpreted as numbers of Motzkin paths which have no horizontal steps at positive heights. Assigning suitable weights gives some nice polynomial extensions. We determine the Hankel determinants and their generating functions for the middle binomial coefficients and derive many conjectures for their polynomial extensions. Finally, we explore experimentally some modifications of the middle binomial coefficients whose Hankel determinants show an interesting modular pattern and obtain some q-analogs.