Shifted Hankel Determinants of Catalan Powers

Abstract

Odd powers of the Catalan generating function determine shifted Hankel determinants with a conjectured closed form.

Definition 1.1 (The Catalan power coefficients).

Lean statement: D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenDefs.catalanPowerCoeff

Formalization. D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenDefs.catalanPowerCoeff (✓ std3).

Source. Repository-derived.

Acknowledgement. Johann Cigler (2023). Some experimental observations about Hankel determinants of convolution powers of Catalan numbers. DOI: 10.48550/arXiv.2308.07642. URL: https://arxiv.org/abs/2308.07642v2.

Commentary.

For a nonnegative integer r and an integer j, define the rational number C_{r,j} to be r/(2j+r) times binom(2j+r,j) when j is nonnegative, and zero when j is negative. Division by zero gives zero. For positive r, these are the coefficients of the r-th power of the Catalan generating function c(x).

Definition 1.2 (The shifted Hankel determinant).

Lean statement: D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenDefs.shiftedHankel

Formalization. D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenDefs.shiftedHankel (✓ std3).

Source. Repository-derived.

Acknowledgement. Johann Cigler (2023). Some experimental observations about Hankel determinants of convolution powers of Catalan numbers. DOI: 10.48550/arXiv.2308.07642. URL: https://arxiv.org/abs/2308.07642v2.

Commentary.

For a nonnegative integer r, an integer shift s and a nonnegative integer N, define D_{r,s}(N) as the determinant of the N by N matrix with entry C_{r,i+j+s} in row i and column j, where both indices range from zero through N minus one. The determinant of the empty matrix is one, and negative coefficient indices contribute zero.

Definition 1.3 (The odd-power determinant formula).

Lean statement: D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenDefs.claim

Formalization. D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenDefs.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. Johann Cigler (2023). Some experimental observations about Hankel determinants of convolution powers of Catalan numbers. DOI: 10.48550/arXiv.2308.07642. URL: https://arxiv.org/abs/2308.07642v2.

Commentary.

Conjecture 11 asserts that for every integer k at least one, every integer m from zero through k+1, and every nonnegative integer n, D_{2k+1,m-k+1}((2k+1)n+k) = (-1)^(kn+binom(k,2)) (2k+1)^m (n+1)^m. The shift m-k+1 is an integer and may be negative.

References

• Truth anchor: D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenDefs.catalanPowerCoeff
• Truth anchor: D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenDefs.claim
• Truth anchor: D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenDefs.shiftedHankel