scholarly journals Error functions, Mordell integrals and an integral analogue of a partial theta function

2017 ◽  
Vol 177 (1) ◽  
pp. 1-37
Author(s):  
Atul Dixit ◽  
Arindam Roy ◽  
Alexandru Zaharescu
Keyword(s):  
Theta Function ◽  
Error Functions ◽  
Partial Theta Function ◽  
Integral Analogue

scholarly journals Hardy–Petrovitch–Hutchinson’s problem and partial theta function

2013 ◽  
Vol 162 (5) ◽  
pp. 825-861 ◽  
Author(s):  
Vladimir Petrov Kostov ◽  
Boris Shapiro
Keyword(s):  
Theta Function ◽  
Partial Theta Function

scholarly journals On a partial theta function and its spectrum

2016 ◽  
Vol 146 (3) ◽  
pp. 609-623 ◽  
Author(s):  
Vladimir Petrov Kostov
Keyword(s):  
Entire Function ◽  
Theta Function ◽  
Local Minimum ◽  
Real Zero ◽  
Positive Real ◽  
The Real ◽  
Real Zeros ◽  
Partial Theta Function

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 .

scholarly journals On the zeros of a partial theta function

2013 ◽  
Vol 137 (8) ◽  
pp. 1018-1030 ◽  
Author(s):  
Vladimir Petrov Kostov
Keyword(s):  
Theta Function ◽  
Partial Theta Function

Asymptotic expansions, partial theta functions, and radial limit differences of mock modular and modular forms

2020 ◽  
pp. 1-10
Author(s):  
Amanda Folsom
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Asymptotic Expansions ◽  
Theta Functions ◽  
Roots Of Unity ◽  
Radial Limit ◽  
Special Values ◽  
Partial Theta Function ◽  
Finite Sums ◽  
Partial Theta Functions

In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as [Formula: see text] tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function [Formula: see text] were established by Ono, Rhoades, and the author, as a special case of a more general result, in which they were realized as special values of a quantum modular form. Our results here are threefold: we realize these radial limit differences as special values of a partial theta function, provide full asymptotic expansions for the partial theta function as [Formula: see text] tends towards roots of unity radially, and explicitly evaluate the partial theta function at roots of unity as simple finite sums of roots of unity.

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