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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Remarks on some integrals and series involving the Stirling numbers and $ \zeta(n)$

Author: Li-Chien Shen
Journal: Trans. Amer. Math. Soc. 347 (1995), 1391-1399
MSC: Primary 11B73; Secondary 11M06, 11Y60, 33C05
MathSciNet review: 1257124
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Abstract | References | Similar Articles | Additional Information

Abstract: From the perspective of the well-known identity

$\displaystyle {}_2{F_1}(a,b;c;1) = \frac{{\Gamma (c)\Gamma (c - a - b)}} {{\Gamma (c - a)\Gamma (c - b)}},$

we clarify the connections between the Stirling numbers $ s_k^n$ and the Riemann zeta function $ \zeta (n)$. As a consequence, certain series and integrals can be evaluated in terms of $ \zeta (n)$ and $ s_k^n$.

References [Enhancements On Off] (What's this?)

  • [1] B. C. Berndt, Ramanujan's notebooks, part I, Springer-Verlag, New York, 1985.
  • [2] D. Borwein and J. M. Borwein, On some intriguing sums involving $ \zeta (4)$, preprint.
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  • [4] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966.
  • [5] E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed., Cambridge Univ. Press, 1958.

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Additional Information

PII: S 0002-9947(1995)1257124-1
Keywords: Riemann zeta function, hypergeometric series, Stirling numbers of the first kind
Article copyright: © Copyright 1995 American Mathematical Society

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