Hankel determinants and continued fractions

Hankel transforms of sequences and their invariance under binomial transforms, Heilermann's formula linking Jacobi continued fractions to weighted Motzkin paths, Catalan-sum Hankel determinants, and Toeplitz–Hessenberg determinants.

TheoremLayman's theorem: the binomial transform preserves the Hankel transform#✓fully proved

For every sequence of integers \((a_n)_{n \ge 0}\), the Hankel transform of the binomial transform of \(a\) equals the Hankel transform of \(a\). Here the binomial transform of \(a\) is the sequence \(b_n = \sum_{i=0}^{n} \binom{n}{i} a_i\), and the Hankel transform of a sequence \(c\) is the sequence whose \(n\)-th term, for \(n \ge 0\), is the determinant of the \((n+1) \times (n+1)\) matrix \((c_{i+j})_{0 \le i, j \le n}\).

theorem layman_binomial_transform_hankel_invariant
    (a : ℕ → ℤ) :
    hankelTransform (binomialTransform a) = hankelTransform a
Sources
Uses 3
Statement usesBinomial transform · Hankel transform
Proof usesBinomial transform as a binomial sum

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