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On a partial theta function and its spectrum

Published online by Cambridge University Press:  19 April 2016

Vladimir Petrov Kostov*
Affiliation:
Laboratoire de Mathématiques, Université de Nice, Parc Valrose, 06108 Nice Cedex 2, France([email protected])

Extract

The bivariate series defines a partial theta function. For fixed q (∣q∣ < 1), θ(q, ·) is an entire function. For q ∈ (–1, 0) the function θ(q, ·) has infinitely many negative and infinitely many positive real zeros. There exists a sequence of values of q tending to –1+ such that has a double real zero (the rest of its real zeros being simple). For k odd (respectively, k even) has a local minimum (respectively, maximum) at , and is the rightmost of the real negative zeros of (respectively, for k sufficiently large is the second from the left of the real negative zeros of ). For k sufficiently large one has . One has and .

MSC classification

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2016 

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