On the power of number-theoretic operations with respect to counting

We investigate function classes ⟨P⟩_f which are defined as the closure of P under the operation f and a set of known closure properties of P, e.g. summation over an exponential range. First, we examine operations f under which P is closed (i.e., ⟨P⟩_f=P) in every relativization. We obtain the following complete characterization of these operations: P is closed under f in every relativization if and only if f is a finite sum of binomial coefficients over constants. Second, we characterize operations f with respect to their power in the counting context in the unrelativized case. For closure properties f of P, we have ⟨P⟩ _f= P. The other end of the range is marked by operations f for which ⟨P⟩_f corresponds to the counting hierarchy. We call these operations counting hard and give general criteria for hardness. For many operations f we show that ⟨P⟩ _f corresponds to some subclass C of the counting hierarchy. This will then imply that P is closed under f if and only if UP=C; and on the other hand f is counting hard if and only if C contains the counting hierarchy