Complementary Euler numbers
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Complementary Euler numbers Takao Komatsu1
© Akadémiai Kiadó, Budapest, Hungary 2017
n(k) (n = 0, 1, . . .) Abstract For an integer k, define poly-Euler numbers of the second kind E by ∞
Lik (1 − e−4t ) (k) t n En = . 4 sinh t n! n=0
n = When k = 1, E numbers defined by
n(1) E
are Euler numbers of the second kind or complimentary Euler ∞
tn t n . E = sinh t n! n=0
Euler numbers of the second kind were introduced as special cases of hypergeometric Euler numbers of the second kind in Komatsu and Zhu (Hypergeometric Euler numbers, 2016, arXiv:1612.06210), so that they would supplement hypergeometric Euler numbers. In this paper, we study generalized Euler numbers of the second kind and give several properties and applications. Keywords Euler numbers · Complementary Euler numbers · Euler numbers of the second kind · Poly-Euler numbers of the second kind · Determinant · Zeta functions
1 Introduction (k)
For an integer k, poly-Euler numbers E n (n = 0, 1, . . .) are defined by ∞
Lik (1 − e−4t ) (k) t n En = 4t cosh t n!
(1.1)
n=0
B 1
Takao Komatsu [email protected] School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
123
T. Komatsu
[7–9], where Lik (z) =
∞ n z nk
(|z| < 1, k ∈ Z)
n=1
(1)
is the kth polylogarithm function. When k = 1, E n = E n are the Euler numbers defined by ∞
tn 1 En . = cosh t n!
(1.2)
n=0
Euler numbers have been extensively studied by many authors (see e.g. [5,7–10] and references therein), in particular, by means of Bernoulli numbers. In [4], for N ≥ 0 hypergeometric Euler numbers E N ,n (n = 0, 1, 2, . . .) are defined by ∞
1 tn E N ,n , = 2 n! 1 F2 (1; N + 1, (2N + 1)/2; t /4)
(1.3)
n=0
where 1 F2 (a; b, c; z) is the hypergeometric function defined by 1 F2 (a; b, c; z) =
∞ n=0
(a)(n) z n . (b)(n) (c)(n) n!
Here (x)(n) is the rising factorial, defined by (x)(n) = x(x + 1) . . . (x + n − 1) (n ≥ 1) with (x)(0) = 1. Note that 1 t 2N /(2N )! . = N −1 2n 2 1 F2 (1; N + 1, (2N + 1)/2; t /4) cosh t − n=0 t /(2n)! When N = 0, E n = E 0,n are the Euler numbers defined in (1.2). The sums of products of hypergeometric Euler numbers can be expressed as for N ≥ 1 and n ≥ 0, n n n n 2N − k N −1,n−k . E N ,k E E N ,i E N ,n−i = 2N i k i=0
k=0
N ,n are the complementary hypergeometric Euler numbers defined by where E ∞
n t 2N +1 /(2N + 1)! N ,n t = E N −1 2n+1 n! sinh t − n=0 t /(2n + 1)! n=0
(1.4)
0,n are the complementary Euler numbers defined by n = E [4, Theorem 4]. When n = 0, E ∞
tn t n . = E sinh t n!
(1.5)
n=0
n are called weighted Bernoulli numbers. On the other hand, the sums of products In [5], E of complementary hypergeometric Euler numbers can be also expressed as n n n n 2N − k + 1 E N ,k E N ,n−k E N ,i E N ,n−i = 2N + 1 i k i=0
[4, Theorem 6].
123
k=0
Complementary Euler numbers
In this paper, we study generalized Euler numbers of the second kind and give several properties and applications.
2 Expressions in terms of the determinants It is known that the Euler
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