  In this paper, we introduce the hypergeometric Euler number as an analogue of
the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We
study several expressions and sums of products of hypergeometric Euler numbers.
We also introduce complementary hypergeometric Euler numbers and give some
characteristic properties. There are strong reasons why these hypergeometric
numbers are important. The hypergeometric numbers have one of the advantages
that yield the natural extensions of determinant expressions of the numbers,
though many kinds of generalizations of the Euler numbers have been considered
by many authors.
