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Integral Confluence of Formal Alternants

Abstract

The first possible coefficient of a rescaled alternant is a Vandermonde determinant times a coefficient determinant.

Theorem 1.1 (The first alternant coefficient).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelConfluence.alternant_coefficients

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelConfluence.alternant_coefficients (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Johann Cigler (2022). Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths. DOI: 10.48550/arXiv.2204.09910. URL: https://arxiv.org/abs/2204.09910v4.

Commentary.

Let R be any commutative ring, let m be a nonnegative integer, let f_i(y) be m formal power series over R, and let c_j be m elements of R, with indices starting at zero. Form the matrix with entry f_i(c_j y). Every coefficient of its determinant at an index below binom(m,2) is zero. The coefficient at index binom(m,2) equals the determinant of the Vandermonde matrix with entry c_i^j times the determinant of the matrix with entry the coefficient of y^j in f_i(y). The formula imposes no distinctness condition on the c_j and requires no division by factorials or differences of parameters.

References