MathDB

Cigler's odd-index Hankel determinant formulas for Catalan convolution powers

From papers

Let C(x)=∑n≥0CnxnC(x)=\sum_{n\geq 0}C_nx^n be the Catalan generating function, let F(x,r)=C(x)rF(x,r)=C(x)^r, and write Hn(F(x,r))H_n(F(x,r)) for the n×nn\times n Hankel determinant of the coefficients of F(x,r)F(x,r). For a positive integer tt, set r=2t+1r=2t+1. Cigler's odd-index conjecture. For positive integers r=2t+1r=2t+1, the following identities hold:

H(2t+1)n(F(x,2t+1))=H(2t+1)n+1(F(x,2t+1))=(−1)tn,H_{(2t+1)n}(F(x,2t+1))=H_{(2t+1)n+1}(F(x,2t+1))=(-1)^{tn}, H(2t+1)n+t+1(F(x,2t+1))=0,H_{(2t+1)n+t+1}(F(x,2t+1))=0, H(2t+1)n+t(F(x,2t+1))=−H(2t+1)n+t+2(F(x,2t+1))=(−1)tn+(t2)((2t+1)(n+1))t−1,H_{(2t+1)n+t}(F(x,2t+1))=-H_{(2t+1)n+t+2}(F(x,2t+1))=(-1)^{tn+\binom{t}{2}}((2t+1)(n+1))^{t-1}, H(2t+1)n−1(F(x,2t+1))+H(2t+1)n+2(F(x,2t))=(−1)tn+1(t−1)(2t+1).H_{(2t+1)n-1}(F(x,2t+1))+H_{(2t+1)n+2}(F(x,2t))=(-1)^{tn+1}(t-1)(2t+1).

These formulas extend the known modular patterns for small convolution powers; the paper states that they remain unproved in general, although some cases are confirmed computationally.

Progress summary

Solved

A 2024 bijective proof settles the intended conjecture, while the version displayed here has a transcription error in its last identity.

Cigler conjectured determinant identities for Catalan convolution powers with r=2t+1r=2t+1; Wang and Xin restated these in 2018. The intended conjecture is now supported by a complete bijective proof, although the supplied final formula differs from Cigler’s original version.

Known results

  • Wang and Xin (2018) verified the relevant conjectures computationally for r≤31r\le 31.
  • Their paper left Conjectures 7 and 8 unproved in general.

2024 bijective proof

Markus Fulmek’s 2024 paper gives a bijective proof using nonintersecting lattice paths, the Lindström–Gessel–Viennot method, and sign-reversing involutions. A 2024 note by Cigler confirms Fulmek’s proofs and gives further computational arguments; a 2025 paper treats extensions. The displayed final identity appears to be a Wang–Xin transcription using F(x,2t)F(x,2t), whereas Cigler’s original formulation uses the odd power in both determinants.

Current status (as of August 2026): Cigler’s intended odd-index conjecture is resolved by Fulmek’s corroborated bijective proof; the literal statement supplied here requires correction in its final identity.

Sources
Sources & referencesView supporting material

Primary source

Ying Wang and Guoce Xin, “Hankel determinants for convolution powers of Catalan numbers”, arXiv:1811.00248 (2018).

catalan-numbersgenerating-functionshankel-determinants

Solutions 1

Counterexample

Counterexample to the fourth identity and source correction

The fourth displayed identity is false as written. Take t=n=1t=n=1, so r=3r=3. Using

[xj]C(x)p=p2j+p(2j+pj),[x^j]C(x)^p=\frac{p}{2j+p}\binom{2j+p}{j},

we obtain

H2(C3)=det⁡(1339)=0H_2(C^3)= \det\begin{pmatrix}1&3\\3&9\end{pmatrix}=0

and

H5(C2)=det⁡(125144225144213251442132429144213242914304213242914304862)=1.H_5(C^2)= \det\begin{pmatrix} 1&2&5&14&42\\ 2&5&14&42&132\\ 5&14&42&132&429\\ 14&42&132&429&1430\\ 42&132&429&1430&4862 \end{pmatrix}=1.

Therefore the printed left-hand side is

H2(C3)+H5(C2)=1,H_2(C^3)+H_5(C^2)=1,

whereas its right-hand side is

(−1)1⋅1+1(1−1)(2⋅1+1)=0.(-1)^{1\cdot1+1}(1-1)(2\cdot1+1)=0.

Thus the fourth identity is false already for t=n=1t=n=1.

This appears to be a transcription error inherited from Wang–Xin’s restatement. Cigler’s original Conjecture 7.2 uses the odd power 2t+12t+1 in both determinants:

H(2t+1)n−1(C2t+1)+H(2t+1)n+2(C2t+1)=(−1)tn+1(t−1)(2t+1).H_{(2t+1)n-1}(C^{2t+1}) + H_{(2t+1)n+2}(C^{2t+1}) = (-1)^{tn+1}(t-1)(2t+1).

At t=n=1t=n=1, H5(C3)=0H_5(C^3)=0, so Cigler’s original version gives 0+0=00+0=0.

Hence this is a counterexample to the literal MathDB/Wang–Xin formula and an identification of the source-corrected statement—not a proof of Cigler’s remaining all-parameter conjecture.

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Samuel Schlesinger ·