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10
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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Cited by 11 (9 self)
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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
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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Cited by 9 (4 self)
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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.
On asymptotics, Stirling numbers, gamma function and polylogs
 Results Math
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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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Cited by 3 (0 self)
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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.
What is the dimension of citation space?
, 2014
"... We adapt and use methods from the causal set approach to quantum gravity to analyse the structure of citation networks from academic papers on the arXiv, supreme court judgements from the US, and patents. We exploit the causal structure of of citation networks to measure the dimension of the Minkow ..."
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Cited by 1 (1 self)
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We adapt and use methods from the causal set approach to quantum gravity to analyse the structure of citation networks from academic papers on the arXiv, supreme court judgements from the US, and patents. We exploit the causal structure of of citation networks to measure the dimension of the Minkowski space in which these directed acyclic graphs can most easily be embedded explicitly taking time into account as one of the dimensions we are measuring. We show that seemingly similar networks have measurably different dimensions. Our interpretation is that a high dimension corresponds to diverse citation behaviour while a low dimension indicates a narrow range of citations in a field.
On the nature of PhaseType Poisson distributions
"... Matrixform Poisson probability distributions were recently introduced as one matrix generalization of Panjer distributions. We show in this paper that under the constraint that their representation is to be nonnegative, they have a physical interpretation as extensions of PH distributions, and we ..."
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Matrixform Poisson probability distributions were recently introduced as one matrix generalization of Panjer distributions. We show in this paper that under the constraint that their representation is to be nonnegative, they have a physical interpretation as extensions of PH distributions, and we name this restricted family Phasetype Poisson. We use our physical interpretation to construct an EM algorithmbased estimation procedure.