Hankel Determinants of Convolution Powers

• The paper establishes that Hankel determinants of convolution powers exhibit periodic vanishing and defined sign patterns, particularly using Catalan numbers as a key example.
• Methodologies include continued fraction expansions, determinant lemmas, and quadratic-transformation techniques to derive explicit evaluations and polynomial interpolants.
• Implications of this research extend to combinatorics, orthogonal polynomials, and moment theory, with potential applications in random matrix theory and related combinatorial sequences.

A Hankel determinant of convolution powers refers to the determinant of Hankel matrices whose entries are given by the convolution powers of a fixed sequence, most classically the Catalan numbers. Research since 2018 reveals a highly structured, deeply algebraic, and still partially conjectural theory for these determinants, characterized by periodic vanishing, distinctive sign patterns, and the emergence of explicit or polynomial evaluations in residue classes. The subject engages combinatorics, orthogonal polynomials, continued fractions, and the algebraic theory of moments, and has recently been generalized to other combinatorial sequences such as Narayana or Motzkin numbers.

1. Fundamental Definitions and Notation

Let $C_n$ denote the $n$th Catalan number: $C_n = \frac{1}{n+1} \binom{2n}{n}, \qquad n \ge 0,$ with generating function

$$C(x) = \sum_{n \ge 0} C_n\,x^n = \frac{1-\sqrt{1-4x}}{2x}.$$

For any integer $k \ge 1$, the $k$-th convolution power, $C_{k, n}$, is defined via the generating function $C(x)^k = \sum_{n \ge 0} C_{k, n}\,x^n$, with closed form

$$C_{k, n} = \frac{k}{2n + k} \binom{2n + k}{n} = \binom{2n + k - 1}{n} - \binom{2n + k - 1}{n - 1}$$

and $C_{k, n} = 0$ for $n$0.

Given integers $n$1 and $n$2, and $n$3, the shifted Hankel determinant is

$n$4

This construction generalizes to moment sequences arising from generating functions with quadratic relations, and to other combinatorial sequences such as Motzkin or Narayana numbers (Cigler, 2023).

2. Closed Forms, Periodicity, and Explicit Evaluations

Systematic computational and theoretical investigations have uncovered a pervasive periodicity and regularity in the sequences $n$5. For small $n$6 and $n$7, and certain index residues, explicit formulas or sign patterns are rigorously established.

Even Convolution Orders

For $n$8, setting $n$9 yields: $C_n = \frac{1}{n+1} \binom{2n}{n}, \qquad n \ge 0,$0 The nonzero determinants appear only at multiples of $C_n = \frac{1}{n+1} \binom{2n}{n}, \qquad n \ge 0,$1 (periodicity $C_n = \frac{1}{n+1} \binom{2n}{n}, \qquad n \ge 0,$2), with "pure sign" values (Cigler, 2023).

Odd Convolution Orders

For $C_n = \frac{1}{n+1} \binom{2n}{n}, \qquad n \ge 0,$3 and $C_n = \frac{1}{n+1} \binom{2n}{n}, \qquad n \ge 0,$4,

$C_n = \frac{1}{n+1} \binom{2n}{n}, \qquad n \ge 0,$5

and otherwise the determinant vanishes except at specified residues (Cigler, 2023).

Examples

• For $C_n = \frac{1}{n+1} \binom{2n}{n}, \qquad n \ge 0,$6, $C_n = \frac{1}{n+1} \binom{2n}{n}, \qquad n \ge 0,$7 cycles as $C_n = \frac{1}{n+1} \binom{2n}{n}, \qquad n \ge 0,$8, period $C_n = \frac{1}{n+1} \binom{2n}{n}, \qquad n \ge 0,$9.
• For $$C(x) = \sum_{n \ge 0} C_n\,x^n = \frac{1-\sqrt{1-4x}}{2x}.$$0, $$C(x) = \sum_{n \ge 0} C_n\,x^n = \frac{1-\sqrt{1-4x}}{2x}.$$1 alternates as $$C(x) = \sum_{n \ge 0} C_n\,x^n = \frac{1-\sqrt{1-4x}}{2x}.$$2; $$C(x) = \sum_{n \ge 0} C_n\,x^n = \frac{1-\sqrt{1-4x}}{2x}.$$3 is $$C(x) = \sum_{n \ge 0} C_n\,x^n = \frac{1-\sqrt{1-4x}}{2x}.$$4 (Cigler, 2023).

3. Polynomiality, Residue Families, and Moment Theory

For each fixed $$C(x) = \sum_{n \ge 0} C_n\,x^n = \frac{1-\sqrt{1-4x}}{2x}.$$5 and admissible residue $$C(x) = \sum_{n \ge 0} C_n\,x^n = \frac{1-\sqrt{1-4x}}{2x}.$$6, define

$$C(x) = \sum_{n \ge 0} C_n\,x^n = \frac{1-\sqrt{1-4x}}{2x}.$$7

and

$$C(x) = \sum_{n \ge 0} C_n\,x^n = \frac{1-\sqrt{1-4x}}{2x}.$$8

Experimental data supports the conjecture that for each residue class the sequence $$C(x) = \sum_{n \ge 0} C_n\,x^n = \frac{1-\sqrt{1-4x}}{2x}.$$9 is a polynomial in $k \ge 1$0, with degree and factorization governed by classical special numbers such as Bernoulli numbers or Fuss–Catalan constants. For example, $k \ge 1$1 (linear), while for higher $k \ge 1$2 the degree increases (Cigler, 2023).

This reflects deep connections to moment theory and continued fractions. The generating functions $k \ge 1$3 have explicit Stieltjes-type continued fraction expansions. The Hankel determinants then correspond to products of finitely many continued fraction coefficients, which, for convolution powers, explains both the periodicity (polynomial, sign, or zero) and the algebraic structure in the residue classes (Cigler, 2023).

4. Generating Functions and Palindromic Properties

For fixed $k \ge 1$4, the ordinary generating function of the Hankel determinants

$k \ge 1$5

is always a rational function in $k \ge 1$6, whose numerator is palindromic or skew-palindromic. For example,

$k \ge 1$7

with higher $k \ge 1$8 yielding analogous rational expressions (Cigler, 2023). This supports the emergence of symmetries and recurrence properties at the level of generating functions.

5. Quadratic Functional Equations and Generalization

Cigler's identities for Hankel determinants of Catalan convolution powers extend to any power series $k \ge 1$9 satisfying a quadratic equation of the form

$k$0

Explicit formulas are given for all $k$1-fold convolutions via shifted Hankel determinants, with zero patterns for initial indices and periodic shift relations for larger $k$2 (Liu et al., 21 Mar 2025). Motzkin and Narayana generating functions fall into this scheme, broadening the theory's scope.

The resulting structure comprises:

• Zero patterns in early indices, depending on $k$3, and the shift,
• Periodic sign shifts in the nonzero range,
• Explicit calculations of the first nonzero determinants via Lucas polynomials or analogous factors (Cigler, 2024, Liu et al., 21 Mar 2025).

6. Proof Techniques and Computational Approaches

Established cases ($k$4 and special shifted cases for higher $k$5) are proved via

• Continued fraction expansions, with explicit correspondence to moment sequences and orthogonal polynomials,
• Dodgson condensation identities,
• Determinant lemmas generalizing Andrews–Wimp,
• Iterated quadratic-transformation methods (Sulanke–Xin), yielding shifted periodic continued fractions and short recurrences for the determinants (Wang et al., 2018, Cigler, 2024).

For higher $k$6 and general shifts, substantial results remain conjectural, but the numerics are compelling (Cigler, 2023, Cigler, 2023).

7. Open Problems and Connections

Key unsolved questions include:

• Proof of full periodic vanishing and sign patterns for all $k$7,
• General proof of the polynomiality of interpolants $k$8,
• Explicit closed-form factorizations for nonzero determinants in all residue classes (Cigler, 2023).

The appearance of Bernoulli numbers and other special structure in leading coefficients suggests links to classical identities for trigonometric expansions and constant terms.

Broader connections exist to the theory of orthogonal polynomials with periodic recurrence coefficients, to random matrix theory (where Catalan moments arise), and to combinatorics of lattice paths, particularly within the context of shifted moment sequences and their vanishing properties (Cigler, 2023, Han, 14 Dec 2025, Cigler, 2018).

Potential generalizations include binomial-coefficient sequences, Motzkin and Narayana numbers, and possibly other quadratic moment sequences, forming the emerging landscape of periodic moment sequence theory within algebraic combinatorics.