Asymptotics of Generalized Partial Theta Functions with a Dirichlet Character

In this paper, we obtain some Hecke-type identities by using two [Formula: see text]-series expansion formulae. And, the identities can also be proved directly in terms of Bailey pairs. In particular, we show that certain partial theta functions and the theta functions can be expressed in terms of Hecke-type identities.

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On the asymptotics of some partial theta functions

The Ramanujan Journal ◽

10.1007/s11139-017-9893-6 ◽

2017 ◽

Vol 45 (3) ◽

pp. 895-907 ◽

Cited By ~ 1

Author(s):

Richard J. McIntosh

Keyword(s):

Theta Functions ◽

Partial Theta Functions

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COMBINATORIAL PROOFS OF CERTAIN IDENTITIES INVOLVING PARTIAL THETA FUNCTIONS

International Journal of Number Theory ◽

10.1142/s1793042110003046 ◽

2010 ◽

Vol 06 (02) ◽

pp. 449-460 ◽

Cited By ~ 8

Author(s):

BYUNGCHAN KIM

Keyword(s):

Theta Functions ◽

Combinatorial Proof ◽

Partial Theta Functions

In this brief note, we give combinatorial proofs of two identities involving partial theta functions. As an application, we prove an identity for the product of partial theta functions, first established by Andrews and Warnaar. We also provide a generalization of the first two identities and give a combinatorial proof of the generalized identities.

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A heuristic guide to evaluating triple-sums

10.46298/hrj.2021.7431 ◽

2021 ◽

Vol Volume 43 - Special... ◽

Author(s):

Eric Mortenson

Keyword(s):

Theta Functions ◽

False Theta Functions ◽

International Audience ◽

Partial Theta Functions

International audience Using a heuristic that relates Appell-Lerch functions to divergent partial theta functions one can expand Hecke-type double-sums in terms of Appell-Lerch functions. We give examples where the heuristic can be used as a guide to evaluate analogous triple-sums in terms of Appell-Lerch functions or false theta functions.

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ON SOME NEW MOCK THETA FUNCTIONS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788718000368 ◽

2018 ◽

Vol 107 (1) ◽

pp. 53-66

Author(s):

NANCY S. S. GU ◽

LI-JUN HAO

Keyword(s):

Theta Functions ◽

Constant Term ◽

Second Order ◽

Sixth Order ◽

Mock Theta Functions ◽

Dual Nature ◽

Partial Theta Functions

In 1991, Andrews and Hickerson established a new Bailey pair and combined it with the constant term method to prove some results related to sixth-order mock theta functions. In this paper, we study how this pair gives rise to new mock theta functions in terms of Appell–Lerch sums. Furthermore, we establish some relations between these new mock theta functions and some second-order mock theta functions. Meanwhile, we obtain an identity between a second-order and a sixth-order mock theta functions. In addition, we provide the mock theta conjectures for these new mock theta functions. Finally, we discuss the dual nature between the new mock theta functions and partial theta functions.

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

International Journal of Number Theory ◽

10.1142/s1793042120400126 ◽

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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