Hankel determinants and Jacobi continued fractions for q-Euler numbers

The q-analogs of Bernoulli and Euler numbers were introduced by Carlitz in 1948. Similar to recent results on the Hankel determinants for the q-Bernoulli numbers established by Chapoton and Zeng, we perform a parallel analysis for the q-Euler numbers. It is shown that the associated orthogonal polynomials for q-Euler numbers are given by a specialization of the big q-Jacobi polynomials, thereby leading to their corresponding Jacobi continued fraction expressions, which eventually serve as a key to our determinant evaluations.