On four new mock theta functions
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In this paper, we first give some representations for four new mock theta functions defined by Andrews \cite{Andrews} and Bringmann, Hikami and Lovejoy \cite{BHL} using divisor sums. Then, some transformation and summation formulae for these functions and corresponding bilateral series are derived as special cases of $_2\psi_2$ series \[\sum_{n=-\infty}^\infty\frac{(a,c;q)_n}{(b,d;q)_n}z^n\] and Ramanujan's sum \[\sum_{n=-\infty}^\infty\frac{(a;q)_n}{(b;q)_n}z^n.\]
Abstract
In this paper, we first give some representations for four new mock theta functions defined by Andrews \cite{Andrews} and Bringmann, Hikami and Lovejoy \cite{BHL} using divisor sums. Then, some transformation and summation formulae for these functions and corresponding bilateral series are derived as special cases of $_2\psi_2$ series \[\sum_{n=-\infty}^\infty\frac{(a,c;q)_n}{(b,d;q)_n}z^n\] and Ramanujan's sum \[\sum_{n=-\infty}^\infty\frac{(a;q)_n}{(b;q)_n}z^n.\]
참고문헌 (13)
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- "2018", ~
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- K. Hikami, "Transformation formula of the"second"order Mock theta function", Lett. Math. Phys, 75(1), pp.93~98 (2006)
- J. Mc Laughlin, "Analytic number theory, modular forms and q-hypergeometric series" (2017), Springer
- Eric Mortenson, "On three third order mock theta functions and Hecke-type double sums", The Ramanujan Journal, 30(2), pp.279~308 (2013), Springer Science and Business Media LLC[DOI]
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