Applications of an explicit formula for the generalized Euler numbers
For real number \(x\) and positive integers \(n,k\), let \(E^{(x)}_{2n}\), \(s(n,k)\), \(T(n,k)\) denote the generalized Euler numbers, Stirling numbers and central factorial numbers, respectively. In the present paper under review, the authors prove an explicit formula for \(E^{(x)}_{2n}\) as follows: \[ E^{(x)}_{2n}=\sum_{i=1}^n\rho(n,i)x^i, \] here \[ \rho(n,k)=(-1)^k\sum_{j=k}^n\frac{(2j)!}{2^jj!}s(j,k)T(n,j). \] By using this formula, they also obtain some interesting identities and congruences involving the higher-order Euler numbers, Stirling numbers, the central factorial numbers and values of the Riemann zeta function.
scientific article; zbMATH DE number 5278768