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The evaluation of Tornheim double sums, Part 1
, 2006
"... We provide an explicit formula for the Tornheim double series in terms of integrals involving the Hurwitz zeta function. We also study the limit when the parameters of the Tornheim sum become natural numbers, and show that in that case it can be expressed in terms of definite integrals of triple pro ..."
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We provide an explicit formula for the Tornheim double series in terms of integrals involving the Hurwitz zeta function. We also study the limit when the parameters of the Tornheim sum become natural numbers, and show that in that case it can be expressed in terms of definite integrals of triple products of Bernoulli polynomials and the Bernoulli function Ak(q):= kζ ′ (1 − k, q).
Contribution to the Theory of the Barnes Function
"... Abstract. This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for highprecision computation of the Barnes gamma function and Glaisher’s constant are also discussed. 1. ..."
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Abstract. This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for highprecision computation of the Barnes gamma function and Glaisher’s constant are also discussed. 1.
A Double Inequality for the Double Gamma Function
"... Abstract. We establish some upper and lower bounds in terms of polygamma functions for the double gamma function (or Barnes G function). The method we use is new, simple and efficient for such inequalities. ..."
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Abstract. We establish some upper and lower bounds in terms of polygamma functions for the double gamma function (or Barnes G function). The method we use is new, simple and efficient for such inequalities.
Journal de Théorie des Nombres de Bordeaux 18 (2006), 113–123
"... Special values of multiple gamma functions ..."
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Fractional Sums and Eulerlike Identities
, 2005
"... Abstract. We introduce a natural definition for sums of the form x∑ f(ν) ν=1 ν=1 when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the Γ function or Euler’s littleknown formula ∑−1/2 ν=1 1 ν = −2 l ..."
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Abstract. We introduce a natural definition for sums of the form x∑ f(ν) ν=1 ν=1 when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the Γ function or Euler’s littleknown formula ∑−1/2 ν=1 1 ν = −2 ln2. Many classical identities like the geometric series and the binomial theorem nicely extend to this more general setting. Sums with a fractional number of terms are closely related to special functions, in particular the Riemann and Hurwitz ζ functions. A number of results about fractional sums can be interpreted as classical infinite sums or products or as limits, including identities like
A NOTE ON SPECIAL VALUES OF LFUNCTIONS
"... Abstract. In this paper, we link the nature of special values of certain Dirichlet Lfunctions to those of multiple gamma values. 1. ..."
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Abstract. In this paper, we link the nature of special values of certain Dirichlet Lfunctions to those of multiple gamma values. 1.
Key Words and Phrases. Riemann zeta function, Hurwitz zeta function, multiple gamma
"... Using functional properties of the Hurwitz zeta function and symbolic derivatives of the trigonometric functions, the function ζ(2n + 1, p/q) is expressed in several ways in terms of other mathematical functions and numbers, including in particular the Glaisher numbers. ..."
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Using functional properties of the Hurwitz zeta function and symbolic derivatives of the trigonometric functions, the function ζ(2n + 1, p/q) is expressed in several ways in terms of other mathematical functions and numbers, including in particular the Glaisher numbers.