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Euler functions $E\sb \alpha(z)$ with complex $\alpha$ and applications. (English) Zbl 0849.11021
Anastassiou, George (ed.) et al., Approximation, probability, and related fields. Proceedings of a conference, Santa Barbara, CA, USA, May 20-22, 1993. New York, NY: Plenum. 127-150 (1994).
In an earlier paper [Appl. Math. Lett. 5, No. 6, 83-88 (1992; Zbl 0768.11010)], the first and last authors, along with {\it M. Leclerc}, introduced Bernoulli numbers and Bernoulli polynomials with arbitrary complex indices. This paper uses a different method to extend the classical Euler polynomials $E_n (x)$ of degree $n$ to functions $E_\alpha (z)$ in which both the index $\alpha$ and the argument $z$ are complex numbers. The key idea is to define $E_\alpha (z)$ by a contour integral taken along a loop around the negative real axis. Many interesting applications are given. For the entire collection see [Zbl 0840.00040].

11B68Bernoulli and Euler numbers and polynomials