Hankel continued fractions and Hankel determinants of the Euler numbers
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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. 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.References
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Additional Information
- Guo-Niu Han
- Affiliation: I.R.M.A., UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, F-67084 Strasbourg, France
- MR Author ID: 272629
- Email: guoniu.han@unistra.fr
- Received by editor(s): May 30, 2019
- Received by editor(s) in revised form: October 9, 2019
- Published electronically: March 16, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 4255-4283
- MSC (2010): Primary 05A05, 05A10, 05A19, 11B68, 11C20, 30B70
- DOI: https://doi.org/10.1090/tran/8031
- MathSciNet review: 4105523