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On a partial theta function and its spectrum
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 146 / Issue 3 / June 2016
- Published online by Cambridge University Press:
- 19 April 2016, pp. 609-623
- Print publication:
- June 2016
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- Article
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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 
.
defines a 
of values of 
has a double real zero 
(the rest of its real zeros being simple). For 
. One has 
and 
.