Hankel determinants for convolution powers of Narayana polynomials

TL;DR

This paper proves and generalizes a conjecture about Hankel determinants related to convolution powers of Narayana polynomials, using a symbolic computation approach to handle complex expressions and provide rigorous proofs.

Contribution

It introduces a new proof and generalization of Cigler's conjecture on Hankel determinants of Narayana polynomial powers, employing a guess-and-prove method with symbolic computation.

Findings

Closed-form expressions for Hankel determinants obtained

Symbolic computation verified complex intermediate steps

Rigorous proofs provided through computer algebra

Abstract

We prove and generalize a conjecture of Johann Cigler on the Hankel determinants of convolution powers of Narayana polynomials. Our method follows a "guess-and-prove" strategy, relying on established techniques involving Hankel continued fractions. While the final forms of our theorems are given by simple closed expressions, the proofs require us to formulate and manage extremely large and intricate explicit expressions at intermediate stages. Most of the technically involved and lengthy formal verifications are carried out using a symbolic computation program, whose code is available on the author's personal webpage for independent verification. We emphasize that our program delivers rigorous symbolic proofs, rather than merely verifying the initial terms.