Efficient computation of some special functions
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We introduce a new algorithm to efficiently compute the functions belonging to a suitable set ℱ defined as follows: f∈ℱ means that f(s,x), s∈ A⊂ℝ being fixed and x>0, has a power series expansion centred at x_0=1 with convergence radius greater or equal than 1; moreover, it satisfies a difference equation of step 1 and the Euler-Maclaurin summation formula can be applied to f. Denoting Euler's function as Γ, we will show, for x>0, that logΓ(x), the digamma function ψ(x), the polygamma functions ψ^(w)(x), w∈ℕ, w≥1, and, for s>1 being fixed, the Hurwitz ζ(s,x)-function and its first partial derivative ∂ζ/∂ s(s,x) are in ℱ. In all these cases the coefficients of the involved power series will depend on the values of ζ(u), u>1, where ζ is Riemann's function. As a by-product, we will also show how to compute efficiently the Dirichlet L-functions L(s,χ) and L^'(s,χ), s> 1, χ being a primitive Dirichlet character, by inserting the reflection formulae of ζ(s,x) and ∂ζ/∂ s(s,x) into the first step of the Fast Fourier Transform algorithm. Moreover, we will obtain some new formulae and algorithms for the Dirichlet β-function and for the Catalan constant G. Finally, we will study the case of the Bateman G-function. In the last section we will also describe some tests that show a performance gain with respect to a standard multiprecision implementation of ζ(s,x) and ∂ζ/∂ s(s,x), s>1, x>0.
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