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A PROPERTY OF A PARTIAL THETA FUNCTION

Abstract : The series θ(q, x) := ∞ j=0 q j(j+1)/2 x j converges for |q| < 1 and defines a partial theta function. For any fixed q ∈ (0, 1) it has infinitely many negative zeros. It is known that for q taking one of the spectral values q 1 , ˜ q 2 ,. .. (where 0.3092493386. .. = ˜ q 1 < ˜ q 2 < · · · < 1, lim j→∞ q j = 1) the function θ(q, ·) has a double zero which is the rightmost of its real zeros (the rest of them being simple). For q = ˜ q j the partial theta function has no multiple real zeros. We prove that: 1) for q ∈ (˜ q j , ˜ q j+1 ] the function θ is a product of a degree 2j real polynomial without real roots and a function of the Laguerre– Pólya class LP − I; 2) for q ∈ C\0, |q| < 1, θ(q, x) = i (1 + x/x i), where −x i are the zeros of θ; 3) for any fixed q ∈ C\0, |q| < 1, the function θ has at most finitely-many multiple zeros; 4) for any q ∈ (−1, 0) the function θ is a product of a real polynomial without real zeros and a function of the Laguerre– Pólya class LP; 5) for any fixed q ∈ C\0, |q| < 1, and for k sufficiently large, the function θ has a zero ζ k close to −q −k. These are all but finitely-many of the zeros of θ.
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Submitted on : Monday, March 21, 2016 - 4:40:31 PM
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Vladimir Kostov. A PROPERTY OF A PARTIAL THETA FUNCTION. Comptes rendus de l'Académie bulgare des Sciences, Bulgarian Academy of Sciences, 2014. ⟨hal-01291214⟩

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