MathDB

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

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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.

References

Primary source

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

catalan-numbersgenerating-functionshankel-determinants
Posted

Progress summary

Refreshed
Claimed progress

A 2024 proof settles the corrected conjecture, but a posted calculation shows that the formula printed here has a false last line, so the statement needs correction.

Cigler’s conjecture, restated by Wang and Xin in 2018, predicts explicit values for Hankel determinants of odd Catalan convolution powers. Wang and Xin checked the formulas computationally for r≤31r\le 31 but left the general case open.

February 2024 bijective proof

Fulmek gave a bijective proof of Cigler’s conjectured identities using nonintersecting lattice paths, reflection, and the Lindström–Gessel–Viennot method. Cigler subsequently supplied an independent proof, corroborating that the intended odd-index conjecture is settled.

Posted attempt

A posted calculation evaluates H2(C3)+H5(C2)=1H_2(C^3)+H_5(C^2)=1 when t=n=1t=n=1, contradicting the printed fourth identity, whose right-hand side is 00. It identifies the likely correction: both determinants should use C2t+1C^{2t+1}; this attempt has not been independently verified.

Current status (as of August 2026): The intended corrected conjecture is supported by corroborated proofs, while the literal statement here is false or misprinted in its final identity.

Sources

Solutions 1

CounterexampleThis solution needs a summarySee full solutionHide full solution

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.

Sources:

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Samuel Schlesinger Posted