Crandall’s computation of the incomplete Gamma function and the Hurwitz zeta function, with applications to Dirichlet L-series

Bailey, David H.; Borwein, Jonathan M.

doi:10.1016/j.amc.2015.06.048

Cited by 12 publications

(17 citation statements)

References 19 publications

(29 reference statements)

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“…This need spurred the late Richard Crandall to compile a set of unified and rapidly convergent algorithms (some new, some gleaned from existing literature) for a variety of special functions, fitted for practical implementation and efficient for very high-precision computation [27]. Crandall's work has been in part described and extended by the current authors in [7]. Since, as we have illustrated, the polylogarithms and their relatives are central to a great deal of mathematics and mathematical physics [4,21,31], such an effort is bound to pay off in the near future.…”

Section: Resultsmentioning

confidence: 99%

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Computation and theory of Mordell–Tornheim–Witten sums II

Bailey

Borwein

2015

Journal of Approximation Theory

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In [8] the current authors, along with the late and much-missed Richard Crandall , considered generalized Mordell-Tornheim-Witten (MTW) zeta-function values along with their derivatives, and explored connections with multiple-zeta values (MZVs). This entailed use of symbolic integration, high precision numerical integration, and some interesting combinatorics and special-function theory. The original motivation was to represent objects such as Eulerian log-gamma integrals; and all such integrals were expressed in terms of a MTW basis. Herein, we extend the research envisaged in [8] by analyzing the relations between a significantly more general class of MTW sums. This has required significantly more subtle scientific computation and concomitant special function theory.

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Section: Resultsmentioning

confidence: 99%

“…These approaches are extended to character polylogarithms in [6]. Some related results and computational algorithms for the incomplete gamma function, the Hurwitz zeta function, and the Dirichlet L-series are presented in [7].…”

Section: Introductionmentioning

confidence: 99%

Computation and theory of Mordell–Tornheim–Witten sums II

Bailey

Borwein

2015

Journal of Approximation Theory

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“…As pointed out in [1] and discussed later (see Section 3), this series proves to be very useful computationally. In this form the series cannot be applied when a = −n, n = 1, 2, .…”

Section: Series Expansionmentioning

confidence: 99%

Algorithm 969

Gil

Ruiz-Antolín

Segura

et al. 2016

ACM Trans. Math. Softw.

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An algorithm for computing the incomplete gamma function γ * (a, z) for real values of the parameter a and negative real values of the argument z is presented. The algorithm combines the use of series expansions, Poincaré-type expansions, uniform asymptotic expansions and recurrence relations, depending on the parameter region. A relative accuracy ∼ 10 −13 in the parameter region (a, z) ∈ [−500, 500]×[−500, 0) can be obtained when computing the function γ * (a, z) with the Fortran 90 module IncgamNEG implementing the algorithm.

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“…where we assume that a > 0 and x ≥ 0, and Γ(a) = ∞ 0 t a−1 exp(−t)dt is the wellknown Gamma function. The problem of inverting functions Q(a, x) and P (a, x) is one of the central problems in statistical analysis and applied probability, with various applications [8,2].…”

Section: The Regularized Upper and Lower Incomplete Gamma Functions Amentioning

confidence: 99%

Efficient procedure to generate generalized Gaussian noise using linear spline tools

Monir

Mraoui²,

Hilali³

2021

bspm

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In this paper, we propose a simple method to generate generalized Gaussian noises using the inverse transform of cumulative distribution. This inverse is expressible by means of the inverse incomplete Gamma function. Since the implementation of Newton's method is rather simple, for approximating inverse incomplete Gamma function, we propose a better and new initial value exploiting the close relationship between the incomplete Gamma function and its piecewise linear interpolant. The numerical results highlight that the proposed method simulates well the univariate and bivariate generalized Gaussian noises.

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Crandall’s computation of the incomplete Gamma function and the Hurwitz zeta function, with applications to Dirichlet L-series

Cited by 12 publications

References 19 publications

Computation and theory of Mordell–Tornheim–Witten sums II

Computation and theory of Mordell–Tornheim–Witten sums II

Algorithm 969

Efficient procedure to generate generalized Gaussian noise using linear spline tools

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