#P-hardness of Computing High Order Derivative and Its Logarithm

In this paper, we show a connection between #P and computing the (real) value of the high order derivative at the origin. Consider, as a problem instance, an integer b and a sufficiently often differentiable function F(x) that is given as a string. Then we consider computing the value F-(b)(0) of the b-th derivative of F(x) at the origin. By showing a polynomial as an example, we show that we have FP = #P if we can compute log(2) F-(b)(0) up to certain precision. The previous statement holds even if F(x) is limited to a function that is analytic at any x is an element of R. It implies the hardness of computing the b-th value of a number sequence from the closed form of its generating function.