Generating functions and generalized Euler numbers

Abstract

In this paper we shall give an explicit formula for the coefficient of the expansion of a given generating function raised to an arbitrary power, when that function has an appropriate form. One of the many examples is the generalized Euler numbers and we shall clarify the situation surrounding the congruence $E_{\frac{p-1}{2}}\not\equiv 0 (\text{mod}\ p), p\equiv 1 (\text{mod}\ 4)$, a prime, and restore the priority. At the same time we shall state the true meaning of such a congruence.