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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-1995-1257124-1
MathSciNet review: 1257124
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Abstract: From the perspective of the well-known identity \[ {}_2{F_1}(a,b;c;1) = \frac {{\Gamma (c)\Gamma (c - a - b)}} {{\Gamma (c - a)\Gamma (c - b)}},\] 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?)

    B. C. Berndt, Ramanujan’s notebooks, part I, Springer-Verlag, New York, 1985. D. Borwein and J. M. Borwein, On some intriguing sums involving $\zeta (4)$, preprint. G. Polya and G. Szegő, Problems and theorems in analysis, Vol. I, Springer-Verlag, New York, 1972. W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966. E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed., Cambridge Univ. Press, 1958.

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Keywords: Riemann zeta function, hypergeometric series, Stirling numbers of the first kind
Article copyright: © Copyright 1995 American Mathematical Society