The -analogs of Bernoulli and Euler numbers were introduced by Carlitz in 1948. Similar to recent results on the Hankel determinants for the -Bernoulli numbers established by Chapoton and Zeng, we perform a parallel analysis for the -Euler numbers. It is shown that the associated orthogonal polynomials for -Euler numbers are given by a specialization of the big -Jacobi polynomials, thereby leading to their corresponding Jacobi continued fraction expressions, which eventually serve as a key to our determinant evaluations.
Revised:
Accepted:
Published online:
Shane Chern 1; Lin Jiu 2
CC-BY 4.0
@article{CRMATH_2024__362_G2_203_0,
author = {Shane Chern and Lin Jiu},
title = {Hankel determinants and {Jacobi} continued fractions for $q${-Euler} numbers},
journal = {Comptes Rendus. Math\'ematique},
pages = {203--216},
publisher = {Acad\'emie des sciences, Paris},
volume = {362},
year = {2024},
doi = {10.5802/crmath.569},
language = {en},
}
Shane Chern; Lin Jiu. Hankel determinants and Jacobi continued fractions for $q$-Euler numbers. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 203-216. doi : 10.5802/crmath.569. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.569/
[1] Some determinants of Bernoulli, Euler and related numbers, Port. Math., Volume 18 (1959), pp. 91-99 | MR | Zbl
[2] The theory of partitions, Cambridge Mathematical Library, Cambridge University Press, 1998, xvi+255 pages
[3] Classical orthogonal polynomials, Polynômes orthogonaux et applications (Bar-le-Duc, 1984) (Lecture Notes in Mathematics), Volume 1171, Springer, 1985, pp. 36-62 | DOI | MR | Zbl
[4] -Bernoulli numbers and polynomials, Duke Math. J., Volume 15 (1948), pp. 987-1000 | MR | Zbl
[5] -Ehrhart polynomials of Gorenstein polytopes, Bernoulli umbra and related Dirichlet series, Mosc. J. Comb. Number Theory, Volume 5 (2015) no. 4, pp. 13-38 | MR | Zbl
[6] Moments of -Jacobi polynomials and -zeta values (2020) | arXiv
[7] Nombres de -Bernoulli–Carlitz et fractions continues, J. Théor. Nombres Bordeaux, Volume 29 (2017) no. 2, pp. 347-368 | DOI | Numdam | MR | Zbl
[8] An introduction to orthogonal polynomials, Mathematics and its Applications, 13, Gordon and Breach Science Publishers, 1978
[9] Orthogonal polynomials and Hankel determinants for certain Bernoulli and Euler polynomials, J. Math. Anal. Appl., Volume 497 (2021) no. 1, 124855, 19 pages | MR | Zbl
[10] Hankel determinants of sequences related to Bernoulli and Euler polynomials, Int. J. Number Theory, Volume 18 (2022) no. 2, pp. 331-359 | DOI | MR | Zbl
[11] Sur les polynômes de Tchebicheff, C. R. Acad. Sci. Paris, Volume 200 (1935), pp. 2052-2053 | Zbl
[12] A propagating algorithm for determining th-order polynomial, least-squares fits, Geophysics, Volume 42 (1977) no. 6, pp. 1265-1276 | DOI
[13] Basic hypergeometric series, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, 2004 | DOI
[14] Ueber die Verwandlung der Reihen in Kettenbrüche, J. Reine Angew. Math., Volume 33 (1846), pp. 174-188 | Zbl
[15] Hypergeometric orthogonal polynomials and their -analogues, Springer Monographs in Mathematics, Springer, 2010, xix+578 pages | DOI
[16] Advanced determinant calculus, Sémin. Lothar. Comb., Volume 42 (1999), B42q, 67 pages | MR | Zbl
[17] Advanced determinant calculus: a complement, Linear Algebra Appl., Volume 411 (2005) no. 6, pp. 68-166 | DOI | MR | Zbl
[18] On the “Favard theorem” and its extensions, J. Comput. Appl. Math., Volume 127 (2001) no. 1-2, pp. 231-254 | DOI | MR | Zbl
[19] A treatise on the theory of determinants, Dover Publications, 1960
[20] Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse, Math., Volume 8 (1894) no. 4, p. J1-J122 | DOI | Numdam | MR
[21] Square Hankel SVD subspace tracking algorithms, Signal Process., Volume 57 (1997) no. 1, pp. 1-18 | DOI | Zbl
[22] Analytic theory of continued fractions, D. Van Nostrand Co., Inc., 1948
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