Some nice Hankel determinants

I study Hankel determinants of a class of sequences which can be interpreted as generalizations of the Catalan numbers and the central binomial coefficients. They follow a modular pattern with a frequent appearance of zeroes, so that the theory of orthogonal polynomials is not applicable. Even so our theorems and conjectures show some similarity with the relations between Catalan numbers and Fibonacci polynomials or between central binomial coefficients and Lucas polynomials.

💡 Research Summary

The paper investigates Hankel determinants of sequences that generalize Catalan numbers and central binomial coefficients. While classical theory links Hankel determinants to orthogonal polynomials, the sequences studied here display a modular pattern with frequent zeros, preventing the direct use of Favard’s three‑term recurrence. The author therefore adopts a generating‑function approach: a formal power series (H_{x,y,z}(z)=\sum_{n\ge0}h_n(x,y,z)z^n) is required to satisfy a quadratic functional equation \