  The hypergeometric zeta function is defined in terms of the zeros of the
Kummer function M(a, a + b; z). It is established that this function is an
entire function of order 1. The classical factorization theorem of Hadamard
gives an expression as an infinite product. This provides linear and quadratic
recurrences for the hypergeometric zeta function. A family of associated
polynomials is characterized as Appell polynomials and the underlying
distribution is given explicitly in terms of the zeros of the associated
hypergeometric function. These properties are also given a probabilistic
interpretation in the framework of Beta distributions.
