Hankel continued fractions and Hankel determinants of the Euler numbers
Hankel continued fractions and Hankel determinants of the Euler numbers
复制标题
欧拉数的汉克尔连分数和汉克尔行列式
DOI：
--
复制
发表时间：
2019
影响因子：
1.3
通讯作者：
Guo
中科院分区：
文献类型：
--
作者：
Guo
欧拉数出现在泰勒展开式中，
谭
(
X
)
+
SEC
(
X
)
an（x）+sec（x）
.自Stieltjes以来，偶数欧拉数的连分数和汉克尔行列式以及奇数欧拉数的连分数和汉克尔行列式已被分别广泛研究。然而，没有汉克尔行列式的（混合）欧拉数已获得。其原因是欧拉数的某些汉克尔行列式为零。这意味着欧拉数的雅可比连分数不存在。本文用Hankel连分式代替Hankel连分式，
J
J
- 分数。因此，欧拉数的汉克尔行列式的显式公式被导出，以及涉及欧拉数的汉克尔连分数和汉克尔行列式的完整列表。最后，新的
Q
Q
欧拉数的类比
E
n
(
Q
)
E_n（q）
基于我们的连分数的。我们得到一个显式公式，
E
n
(
−
1
)
E_n（-1）
证明了R的一个猜想。J. Mathar在这些数字上。
The Euler numbers occur in the Taylor expansion of
tan
(
x
)
+
sec
(
x
)
an (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
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
q
-analog of the Euler numbers
E
n
(
q
)
E_n(q)
based on our continued fraction is proposed. We obtain an explicit formula for
E
n
(
−
1
)
E_n(-1)
and prove a conjecture by R. J. Mathar on these numbers.