IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Online ISSN : 1745-1337
Print ISSN : 0916-8508
Special Section on Discrete Mathematics and Its Applications
#P-hardness of Computing High Order Derivative and Its Logarithm
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2014 Volume E97.A Issue 6 Pages 1382-1384


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 2F(b)(0) up to certain precision. The previous statement holds even if F(x) is limited to a function that is analytic at any xR. It implies the hardness of computing the b-th value of a number sequence from the closed form of its generating function.

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© 2014 The Institute of Electronics, Information and Communication Engineers
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