Hypergeometric summation representations of the Stieltjes constants

Details Hypergeometric summation representations of the Stieltjes constants Hypergeometric summation representations of the Stieltjes constants

2011-06-25

arxiv.org/abs/1106.5148v1

Physics

Mathematical Physics

25 pages, no figures

Scientific paper

The Stieltjes constants $\gamma_k$ appear in the regular part of the Laurent expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and specialized hypergeometric functions $_pF_{p+1}$. This family of results generalizes a representation of the Euler constant in terms of a summation over values of the trigonometric integrals Si or Ci. The series representations are suitable for acceleration. As byproducts, we evaluate certain sine-logarithm integrals and present the leading asymptotic form of the particular $_pF_{p+1}$ functions.

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