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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.
www.elsevier.com/locate/jnt The evaluation of Tornheim double sums, Part 1
, 2004
"... Communicated by D. Goss 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 ..."
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Communicated by D. Goss 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).
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
THE EVALUATION OF TORNHEIM DOUBLE SUMS. PART 1 (1.1)
, 2005
"... Abstract. 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 ..."
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Abstract. 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). The function T(a, b, c) =