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Bulletin of the London Mathematical Society 2005 37(1):15-24; doi:10.1112/S0024609304003625
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© London Mathematical Society

Binomial Sums and Functions of Exponential Type

Javad Mashreghi and Thomas Ransford

Département de mathématiques et de statistique, Université Laval Québec (QC), Canada G1K 7P4; javad.mashreghi{at}mat.ulaval.ca, ransford{at}mat.ulaval.ca

Received 5 May 2003. Revision received 2 February 2004.

Let (an)n≥0 be a sequence of complex numbers, and, for n≥0, let


Formula

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 Formula for all isin > 0, then an=0 for all n > p. Also, given 0 < p < 1, then Formula for all isin > 0 if and only if Formula. It is further shown that, given rß > 1, if bn,cn=O(rßn), then an=O({alpha}n), where Formula, 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 e{phi}(X), where X is a Poisson random variable. 2000 Mathematics Subject Classification 05A10 (primary), 30D15, 46H05, 60E15 (secondary).

First author's research partially supported by grants from NSERC (Canada) and FQNRT (Québec).

Second author's research partially supported by grants from NSERC (Canada), FQNRT (Québec) and the Canada Research Chairs Program.


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