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On asymptotics, Stirling numbers, Gamma function and
"... We apply the Euler–Maclaurin formula to find the asymptotic expansion of the sums � n ..."
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We apply the Euler–Maclaurin formula to find the asymptotic expansion of the sums � n
NEW PROPERTIES OF rSTIRLING SERIES
"... Abstract. The summation of some series involving the Stirling numbers of the first kind can be found in several works but there is no such a computation for Stirling numbers of the second kind let alone the rStirlings. We offer a comprehensive survey and prove new results. 1. ..."
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Abstract. The summation of some series involving the Stirling numbers of the first kind can be found in several works but there is no such a computation for Stirling numbers of the second kind let alone the rStirlings. We offer a comprehensive survey and prove new results. 1.
Asymptotics of the Stirling numbers of the first kind revisited: A saddle point approach
, 2010
"... Using the saddle [ point] method, we obtain from the generating function of the Stirling numbers n of the first kind and Cauchy’s integral formula, asymptotic results in central and noncentral j regions. In the central region, we revisit the celebrated Goncharov theorem with more precision. In the ..."
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Using the saddle [ point] method, we obtain from the generating function of the Stirling numbers n of the first kind and Cauchy’s integral formula, asymptotic results in central and noncentral j regions. In the central region, we revisit the celebrated Goncharov theorem with more precision. In the region j = n − nα [] n, α> 1/2, we analyze the dependence of on α.
Series representations for the Stieltjes constants
, 2009
"... The Stieltjes constants γk(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about s = 1. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives a ..."
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The Stieltjes constants γk(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about s = 1. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of all orders of the Stieltjes coefficients are given. Many other results are obtained, including instances of an exponentially fast converging series representation for γk = γk(1). Some extensions are briefly described, as well as the relevance to expansions of Dirichlet L functions. Key words and phrases Stieltjes constants, Riemann zeta function, Hurwitz zeta function, Laurent expansion, Stirling numbers of the first kind, Dirichlet L functions, Lerch zeta function 2000 AMS codes