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Hypergeometric summation representations of the Stieltjes constants
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Mark W. Coffey
Published/Copyright:
June 1, 2013
Abstract
The Stieltjes constants γk appear in the regular part of the Laurent expansion of the Riemann and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and specialized hypergeometric functions pFp+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 pFp+1 functions.
Keywords: Riemann zeta function; Stieltjes constants; generalized hypergeometric function; Gamma function; digamma function
Published Online: 2013-06
Published in Print: 2013-06
© by Oldenbourg Wissenschaftsverlag, München, Germany
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Articles in the same Issue
- Optimal initial value conditions for local strong solutions of the Navier–Stokes equations in exterior domains
- Hypergeometric summation representations of the Stieltjes constants
- Meromorphic functions whose certain differential polynomials share a small function with finite weight
- Convolutions of slanted half-plane harmonic mappings
- Meromorphic functions that share one small function DM with their first derivative
- A product theorem for the Euler and the Natarajan methods of summability
- Hermite–Hadamard type integral inequalities for geometric-arithmetically s-convex functions
Keywords for this article
Riemann zeta function;
Stieltjes constants;
generalized hypergeometric function;
Gamma function;
digamma function
Articles in the same Issue
- Optimal initial value conditions for local strong solutions of the Navier–Stokes equations in exterior domains
- Hypergeometric summation representations of the Stieltjes constants
- Meromorphic functions whose certain differential polynomials share a small function with finite weight
- Convolutions of slanted half-plane harmonic mappings
- Meromorphic functions that share one small function DM with their first derivative
- A product theorem for the Euler and the Natarajan methods of summability
- Hermite–Hadamard type integral inequalities for geometric-arithmetically s-convex functions
