[Paper Review] A q-analogue of Catalan Hankel determinants

Masao Ishikawa, Hiroyuki Tagawa+1 more|arXiv (Cornell University)|Sep 10, 2010

Advanced Combinatorial Mathematics15 references8 citations

TL;DR

This paper presents a q-analogue of Catalan Hankel determinants using the q-hypergeometric sequence $\mu_n = \frac{(aq;q)_n}{(abq^2;q)_n}$, establishing a closed-form formula for the Hankel determinant $\det(\mu_{i+j})_{0\leq i,j\leq n-1}$ via lattice path enumeration, orthogonal polynomials, and basic hypergeometric series. The key result generalizes classical Catalan determinant identities to the q-setting with explicit product formulas involving q-Pochhammer symbols and combinatorial weights.

ABSTRACT

In this paper we shall survey the various methods of evaluating Hankel determinants and as an illustration we evaluate some Hankel determinants of a q-analogue of Catalan numbers. Here we consider $\frac{(aq;q)_{n}}{(abq^{2};q)_{n}}$ as a q-analogue of Catalan numbers $C_{n}=\frac1{n+1}\binom{2n}{n}$, which is known as the moments of the little q-Jacobi polynomials. We also give several proofs of this q-analogue, in which we use lattice paths, the orthogonal polynomials, or the basic hypergeometric series. We also consider a q-analogue of Schröder Hankel determinants, and give a new proof of Moztkin Hankel determinants using an addition formula for ${}_2F_{1}$.

Motivation & Objective

• To extend classical Catalan Hankel determinant identities to a q-analogue using a q-hypergeometric sequence.
• To provide multiple proofs of the q-analogue determinant formula using lattice paths, orthogonal polynomials, and basic hypergeometric series.
• To generalize the result to shifted Hankel determinants $\det(\mu_{i+j+t})$ for non-negative integers $t$.
• To explore connections between q-analogues of Catalan numbers and orthogonal polynomials, particularly the little q-Jacobi polynomials.
• To establish q-analogues of Schröder and Motzkin Hankel determinants using similar techniques.

Proposed method

• Define the q-analogue of Catalan numbers as $\mu_n = \frac{(aq;q)_n}{(abq^2;q)_n}$, which arises as moments of the little q-Jacobi polynomials.
• Use the Lindström-Gessel-Viennot lemma to evaluate the Hankel determinant via non-intersecting lattice paths with weighted steps.
• Apply the theory of orthogonal polynomials and continued fractions to derive determinant identities.
• Employ basic hypergeometric series identities, particularly those involving $_2\phi_1$ and $_2F_1$, to prove determinant evaluations.
• Use the Desnanot-Jacobi adjoint matrix theorem to derive recurrence relations for Hankel determinants of q-analogues.
• Establish a q-analogue of the Motzkin Hankel determinant using an addition formula for $_2F_1$.

Experimental results

Research questions

• RQ1 What is the q-analogue of the classical Hankel determinant $\det(C_{i+j})$ for Catalan numbers?
• RQ2 How can the determinant $\det(\mu_{i+j})$ be evaluated in closed form for the q-analogue $\mu_n = \frac{(aq;q)_n}{(abq^2;q)_n}$?
• RQ3 Can multiple proofs—combinatorial (lattice paths), algebraic (orthogonal polynomials), and hypergeometric (basic series)—be unified for the same determinant identity?
• RQ4 What is the structure of the Hankel determinant for shifted indices $\mu_{i+j+t}$, and how does it generalize the unshifted case?
• RQ5 How do q-analogues of Schröder and Motzkin numbers relate to Hankel determinants, and can their determinant identities be derived using similar methods?

Key findings

• The Hankel determinant $\det(\mu_{i+j})_{0\leq i,j\leq n-1}$ is given by $a^{n(n-1)/2} q^{n(n-1)(2n-1)/6} \prod_{k=1}^n \frac{(q,aq,bq;q)_{n-k}}{(abq^{n-k+1};q)_{n-k}(abq^2;q)_{2(n-k)}}$.
• For shifted indices, $\det(\mu_{i+j+t})_{0\leq i,j\leq n-1}$ is expressed as a product involving $\left(\frac{(aq;q)_t}{(abq^2;q)_t}\right)^n$ and additional q-Pochhammer terms.
• The q-analogue of the Motzkin Hankel determinant is proven using an addition formula for $_2F_1$, yielding a determinant value of 1 for $\det(M_{i+j})$.
• The large Schröder Hankel determinant $\det(S_{i+j})$ is shown to equal $2^{n(n-1)/2}$, and $\det(S_{i+j+1}) = 2^{n(n+1)/2}$, with a new formula for $\det(S_{i+j+2})$ derived.
• A conjectured identity for Delannoy number Hankel determinants is proposed: $\det(D(i+j,i+j)) = 2^{\binom{n+1}{2} - 1}$, with similar formulas for higher shifts.
• The paper establishes that the determinant $\det S^{(0)}_n(1)$ satisfies the recurrence $\det S^{(0)}_n(1) = 2^{n(n+1)/2} - (2n+3)2^{n(n-1)/2} - 2^{(n-1)n/2}$, consistent with known values.