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Transformation formula of the “second order” mock theta function. (English) Zbl 1109.33020

Summary: We give a transformation formula for the “second order” mock theta function \[ D_5(q) = \sum_{n=0}^{\infty}\frac{(-q)_n}{(q;q^2)_{n+1}}q^n \] which was recently proposed in connection with the quantum invariant for the Seifert manifold

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
11F27 Theta series; Weil representation; theta correspondences
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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[1] Andrews G.E. (1966). q-identities of Auluck, Carlitz, and Rogers. Duke Math. J. 33:575–582 · Zbl 0152.05202
[2] Andrews G.E. (1981). Mordell integrals and Ramanujan’s ”lost” notebook. In: Knopp M.I (eds). Analytic number theory, Lecture notes in mathematics vol 899. Springer, Berlin Heidelberg New York, pp. 10–48 · Zbl 0482.33002
[3] Andrews G.E., Berndt B.C. (2005). Ramanujan’s lost notebook part I. Springer, Berlin Heidelberg New York · Zbl 1075.11001
[4] Fine N.J. (1988). Basic hypergeometric series and applications. Mathematical survey monographs no. 27, AMS, Providence
[5] Hardy G.H., Aiyar P.V.S., Wilson B.M (eds). (2000). Collected papers of Srinivasa Ramanujan. American Mathematical Society, Providence
[6] Hikami, K.: On the quantum invariant for the spherical Seifert manifold, preprint (2005) math-ph/0504082
[7] Hikami K. (2005). Mock (false) theta functions as quantum invariants. Regular & Chaotic Dynamics 10:509–530 · Zbl 1133.57301
[8] Lawrence R., Zagier D. (1999). Modular forms and quantum invariants of 3-manifolds. Asian J Math 3:93–107 · Zbl 1024.11028
[9] Milnor J. (1975). On the 3-dimensional Brieskorn manifolds M(p,q,r). In: Neuwirth L.P (eds). Knots, groups, and 3-manifolds. Princeton University Press, Princeton, pp. 175–225 papers Dedicated to the Memory of R. H. Fox · Zbl 0305.57003
[10] Montesinos J.M. (1987). Classical tessellations and three-manifolds. Springer, Berlin Heidelberg New York · Zbl 0626.57002
[11] Mordell L.J. (1933). The definite integral \(\int_{-\infty}^\infty \frac{e^{a x^2+bx}}{e^{cx}+d} dx\) and the analytic theory of numbers. Acta Math. 61:323–360 · Zbl 0008.05501
[12] Ramanujan S. (1987). The lost notebook and other unpublished papers. Narosa, New Delhi · Zbl 0639.01023
[13] Reshetikhin N.Yu., Turaev V.G. (1991). Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103:547–597 · Zbl 0725.57007
[14] Watson G.N. (1936). The final problem: an account of the mock theta functions. J. London Math. Soc. 11:55–80 · Zbl 0013.11502
[15] Witten E. (1989). Quantum field theory and Jones’ polynomial. Commun. Math. Phys. 121:351–399 · Zbl 0667.57005
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