Poly-Euler number

In mathematics, poly-Euler numbers, denoted as E_n^(k), and their generalizations were defined by Yoshitaka Sasaki, Abdelmejid Bayad and Hassan Jolany as

$${2\operatorname{Li}_k(1-e^{-x}) \over 1+e^{-x}}=\sum_{n=0}^\infty E_n^{(k)}{x^n\over n!}$$

where Li is the polylogarithm. The E_n⁽¹⁾ are the usual Euler numbers.

The generalization of Poly-Euler numbers with a,b,c parameters defined by Hassan Jolany as

$${2\operatorname{Li}_k(1-(ab)^{-x})\over a^{-x}+b^x}c^{xt}=\sum_{n=0}^\infty E_n^{(k)}(t;a,b,c){x^n \over n!}$$

where Li is the polylogarithm.

Poly-Euler numbers have the same duality which known as Poly-Bernoulli numbers

References

• Hassan Jolany, Mohsen Aliabadi, Roberto B. Corcino, and M.R.Darafsheh, A Note On Multi Poly-Euler Numbers And Bernoulli Polynomials, ,2012
• Hassan Jolany and Roberto B. Corcino, More Properties on Multi Poly-Euler Polynomials, 1
• Hassan Jolany, Explicit formula for generalization of Poly-Bernoulli numbers and polynomials with a,b,c parameters, 2