Hankel Determinants for Convolution of Power Series: An Extension of Cigler's Results

TL;DR

The paper extends Cigler's shifted Hankel-determinant identities from Catalan numbers to the convolution powers of a general power series $F(x)$ that satisfies a quadratic equation. It develops a determinant framework with $[D(x,y)]_n$ and a key lemma, then proves a general formula for $F(x)^k$ in terms of polynomials $L_k(x)$ and derives shift-vanishing identities for $D_{K,M}(N)$ when $F$ satisfies a specific quadratic form. Applications to Catalan and Motzkin generating functions yield explicit $L_k(x)$, degree bounds, and corresponding determinant identities, including connections to Chebyshev and Lucas polynomials and a Sulanke-Xin based analysis that reveals shifted-periodic patterns in exact determinant values. The work provides concrete closed-form determinant sequences for particular $(K,M)$, demonstrates a method to compute them via quadratic transformations, and raises the open problem of a bijective proof for Motzkin-related corollaries. These results deepen the understanding of Hankel determinants in combinatorics and their algebraic structure under quadratic constraints.

Abstract

Cigler considered certain shifted Hankel determinants of convolution powers of Catalan numbers and conjectured identities for these determinants. Recently, Fulmek gave a bijective proof of Cigler's conjecture. Cigler then provided a computational proof. We extend Cigler's determinant identities to the convolution of general power series $F(x)$, where $F(x)$ satisfies a certain type of quadratic equation. As an application, we present the Hankel determinant identities of convolution powers of Motzkin numbers.

Theorems & Definitions (19)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• Remark 2.3
• proof : First proof of Theorem \ref{['FK-convlution']}
• proof : Second proof of Theorem \ref{['FK-convlution']}
• Theorem 2.4
• proof
• Corollary 3.1