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Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series

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Analytic Number Theory, Modular Forms and q-Hypergeometric Series (ALLADI60 2016)

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Abstract

The bilateral series corresponding to many of the third-, fifth-, sixth-, and eighth-order mock theta functions may be derived as special cases of \(_2\psi _2\) series

$$ \sum _{n=-\infty }^{\infty }\frac{(a,c;q)_n}{(b,d;q)_n}z^n. $$

Three transformation formulae for this series due to Bailey are used to derive various transformation and summation formulae for both these mock theta functions and the corresponding bilateral series. New and existing summation formulae for these bilateral series are also used to make explicit in a number of cases the fact that for a mock theta function, say \(\chi (q)\), and a root of unity in a certain class, say \(\zeta \), that there is a theta function \(\theta _{\chi }(q)\) such that

$$ \lim _{q \rightarrow \zeta }(\chi (q) - \theta _{\chi }(q)) $$

exists, as \(q \rightarrow \zeta \) from within the unit circle.

This paper is dedicated to Krishna Alladi on the occasion of his 60th birthday

This work was partially supported by a grant from the Simons Foundation (#209175 to James Mc Laughlin).

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Authors and Affiliations

  1. Mathematics Department, West Chester University, 25 University Avenue, West Chester, PA, 19383, USA

    James Mc Laughlin

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  1. James Mc Laughlin
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Correspondence to James Mc Laughlin .

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Editors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, State College, Pennsylvania, USA

    George E. Andrews

  2. Department of Mathematics, University of Florida, Gainesville, Florida, USA

    Frank Garvan

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Mc Laughlin, J. (2017). Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series. In: Andrews, G., Garvan, F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham. https://doi.org/10.1007/978-3-319-68376-8_29

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