Hankel Continued fractions and Hankel determinants of the Euler numbers

TL;DR

This paper develops Hankel continued fractions for Euler numbers, deriving explicit formulas for their Hankel determinants and introducing a new q-analog, overcoming previous obstacles caused by null determinants.

Contribution

It introduces Hankel continued fractions for Euler numbers, providing explicit formulas for their determinants and proposing a novel q-analog based on these continued fractions.

Findings

Explicit formulas for Euler number Hankel determinants

Full list of Hankel continued fractions involving Euler numbers

A new q-analog of Euler numbers and proof of Mathar's conjecture

Abstract

The Euler numbers occur in the Taylor expansion of $\tan(x)+\sec(x)$. Since Stieltjes, continued fractions and Hankel determinants of the even Euler numbers, on the one hand, of the odd Euler numbers, on the other hand, have been widely studied separately. However, no Hankel determinants of the (mixed) Euler numbers have been obtained and explicitly calculated. The reason for that is that some Hankel determinants of the Euler numbers are null. This implies that the Jacobi continued fraction of the Euler numbers does not exist. In the present paper, this obstacle is bypassed by using the Hankel continued fraction, instead of the $J$-fraction. Consequently, an explicit formula for the Hankel determinants of the Euler numbers is being derived, as well as a full list of Hankel continued fractions and Hankel determinants involving Euler numbers. Finally, a new $q$-analog of the Euler numbers $E_n(q)$ based on our continued fraction is proposed. We obtain an explicit formula for $E_n(-1)$ and prove a conjecture by R. J. Mathar on these numbers.