On the generalized Euler polynomials of the second kind (Q2855144)

The authors define generalized Euler numbers and polynomials of the second kind as follows: NEWLINE\[NEWLINE\left(\frac{2e^t}{e^{2t}+1}\right)^x=\sum_{n=0}^\infty \tilde{\mathcal E_n}(x)\frac{t^n}{n!}\qquad (| t|<\tfrac {\pi}2;\;1^x:=1),NEWLINE\]NEWLINENEWLINEwhere \(x\) be a real or complex parameter and \(\tilde{\mathcal E}_n(x)\) are called \(n\)th generalized Euler polynomials of the second kind. In the special case \(x=1\) \(\tilde{\mathcal E}_n(1)=\tilde{\mathcal E}_n\) are called \(n\)th generalized Euler numbers of the second kind. NEWLINENEWLINEIn this paper the authors obtain some identities and relations between generalized Euler polynomials of the second kind and Bernoulli numbers, Euler numbers and Stirling numbers.