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Applications of an explicit formula for the generalized Euler numbers. (English) Zbl 1145.11020
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.

11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
11M38 Zeta and \(L\)-functions in characteristic \(p\)
11B83 Special sequences and polynomials
05A10 Factorials, binomial coefficients, combinatorial functions
Full Text: DOI
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