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Binomial Sums and Functions of Exponential Type

Binomial Sums and Functions of Exponential Type Abstract Let (an)n≥0 be a sequence of complex numbers, and, for n≥0, let   bn=∑k=0n(nk)akandcn=∑k=0n(nk)(−1)n−kak A number of results are proved relating the growth of the sequences (bn) and (cn) to that of (an). For example, given p≥0, if bn = O(np and cn=O(e∈n) for all ∈ > 0, then an=0 for all n > p. Also, given 0 < p < 1, then bn,cn=O(e∈np) for all ∈ > 0 if and only if n1/p−1|an|1/n→0. It is further shown that, given rβ > 1, if bn,cn=O(rβn), then an=O(αn), where α=rβ2−1, thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredients of the proogs are a Phragmén-Lindelöf theorem for entire functions of exponential type zero, and an estimate for the expected value of eϕ(X), where X is a Poisson random variable. 2000 Mathematics Subject Classification 05A10 (primary), 30D15, 46H05, 60E15 (secondary). © London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Binomial Sums and Functions of Exponential Type

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References (7)

Publisher
Oxford University Press
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/S0024609304003625
Publisher site
See Article on Publisher Site

Abstract

Abstract Let (an)n≥0 be a sequence of complex numbers, and, for n≥0, let   bn=∑k=0n(nk)akandcn=∑k=0n(nk)(−1)n−kak A number of results are proved relating the growth of the sequences (bn) and (cn) to that of (an). For example, given p≥0, if bn = O(np and cn=O(e∈n) for all ∈ > 0, then an=0 for all n > p. Also, given 0 < p < 1, then bn,cn=O(e∈np) for all ∈ > 0 if and only if n1/p−1|an|1/n→0. It is further shown that, given rβ > 1, if bn,cn=O(rβn), then an=O(αn), where α=rβ2−1, thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredients of the proogs are a Phragmén-Lindelöf theorem for entire functions of exponential type zero, and an estimate for the expected value of eϕ(X), where X is a Poisson random variable. 2000 Mathematics Subject Classification 05A10 (primary), 30D15, 46H05, 60E15 (secondary). © London Mathematical Society

Journal

Bulletin of the London Mathematical Society – Oxford University Press

Published: Feb 1, 2005

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