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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.

##### MSC:
 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
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